The FR-BDF Model and an Assessment of Monetary Policy

Transmission in France

Matthieu Lemoine1,2, Harri Turunen1, Mohammed Chahad1, Antoine Lepetit1, Anastasia Zhutova1, Pierre

Aldama1, Pierrick Clerc1 & Jean-Pierre Laffargue3

October 2019, WP #736

ABSTRACT

This paper presents the new model for France of the Banque de France (FR-BDF), as well as its key implications for the analysis of monetary policy transmission in France. Relative to our former model, this new semi-structural model has been improved along three dimensions: financial channels are richer, expectations now have an explicit role and simulations now converge toward a balanced growth path. We follow the approach of the FRB/US model, where agents can form their expectations in two different ways, VAR-based or model-consistent, and where non-financial behavior react with polynomial adjustment costs. For standard monetary policy shocks, FR-BDF shows a stronger sensitivity than our former model, due to the widespread influence of expectations. Then, we show that, under model-consistent expectations, FR-BDF does not suffer from the forward guidance puzzle. Finally, Eurosystem asset purchase programmes have notable effects in FR-BDF, with a stronger transmission through exchange rates than term premia.

Keywords: semi-structural modeling, expectations, monetary policy, forward guidance

JEL classification: C54, E37

1 Banque de France 2 Corresponding author, matthieu.lemoine@banque-france.fr 3 University of Paris 1

Authors are listed according to the duration of their participation in the project, but authors who participated for similar durations are listed alphabetically (M. Chahad, A. Lepetit and A. Zhutova). We would particularly like to thank Jean-François Ouvrard for his continuous-time feedback on this paper and on the whole modeling project behind it. We also have special thanks for Stéphane Dees, Yannick Kalantzis, Pierre Sicsic and Camille Thubin for their careful reread and detailed comments. We also thank Werner Roeger for useful discussion and comments. This paper is the outcome of a modeling project of the Banque de France with a well-defined governance and many contributors, which we expand on in the special acknowledgment.

Working Papers reflect the opinions of the authors and do not necessarily express the views of the Banque de France. This document is available on publications.banque-france.fr/en

Banque de France WP #736 ii

NON-TECHNICAL SUMMARY

The model for France of the Banque de France (FR-BDF) is the new semi-structural

replacement of the older model Mascotte. FR-BDF is a large-scale model for France, which

contains detailed behavioral equations as well as a detailed accounting framework. It is used

both for medium-run projection exercises and for policy analysis. The French economy is

modeled as a small-open economy under fixed exchange rates with an exogenous interest rate

due to the constraints of the Eurosystem projection framework. The overall model structure is

strongly inspired by the US model of the Federal Reserve Board (FRB/US). This model combines

economically meaningful expectations, a good empirical fit due to the Polynomial Adjustment

Costs (PAC) framework and ease of estimation due to the separability of model blocks.

Three key features should be emphasized among the improvements of FR-BDF with respect to

Mascotte. First, FR-BDF has richer financial channels than Mascotte. It has a large set of

interest rates, an endogenous term structure and endogenous nominal exchange rates.

Second, expectations play an explicit role, both for financial and non-financial variables, as

expectations are a major transmission channel of monetary policy shocks. Expectations can be

modeled based on a vector autoregressive model (VAR) which summarizes the state of the

economy. We can also allow expectations to be consistent with the model and forward-

looking, which we label Model-Consistent Expectations (MCE). Third, FR-BDF has a well-

defined supply block as well as a balanced growth path toward which it converges smoothly

and endogenously in unconditional simulations.

Concerning short run dynamics of FR-BDF under VAR-based expectations, one striking feature

of our impulse responses to positive and temporary demand shocks is related to the absence

of any feedback from monetary policy. The expansionary effect of these shocks deteriorates

the competitiveness, weights on the output gap, leading later to disinflation and bringing the

real exchange rate back to its baseline level. Impulse responses to supply shocks illustrate the

important role played by the supply block of this model at horizons that could matter for

medium-run forecasts. For example, after a shock that gradually raises labor efficiency by 1%,

real output increases by 0.6 percentage points over a 4-year horizon.

In a last section, we analyze monetary policy transmission in France with three experiments

through the lens of FR-BDF. First, we assess the impact of a standard monetary policy shock.

FR-BDF shows a stronger response than our former model and this can be related to the

influence of the short term rate through expectations. There are some differences in outcomes

that depend on how expectations are modelled (see figure): when non-financial variables are

modeled in a forward-looking manner (full MCE case), this creates a dampening effect in

comparison to the case where they would be backward looking (hybrid case).

Second, we simulate FR-BDF with model-consistent expectations in order to assess the impact

in France of forward guidance in the form of an announced cut in short-term interest rate for a

varying amount of quarters. It appears that FR-BDF does not suffer from the forward guidance

puzzle, i.e. the impact of forward guidance increases linearly and not exponentially with the

length of the shock, thanks to the small sensitivity of consumption to interest rates and to

discounting schemes used in consumption and term structure equations.

Banque de France WP #736 iii

The final experiment we consider deals with how Eurosystem Asset Purchase Programmes (APP) have affected the French economy between 2015 and 2018. We only take into account at this stage direct effects on the French economy and do not those coming from the rest of the euro area. Our main results show that APP had notable real and nominal effects on the French economy, with exchange rates as a stronger channel of transmission than term premia.

Monetary policy responses under different types of expectations

Note: responses for VAR-based, hybrid (Hyb.E) and model-consistent expectations (MCE). Hybrid expectations mix VAR-based expectations for non-financial variables and MCE for financial ones.

Le modèle FR-BDF et une évaluation des effets de la politique monétaire en France

RÉSUMÉ

Cet article présente le nouveau modèle pour la France de la Banque de France (FR-BDF), ainsi que ses implications pour l’analyse de la transmission de la politique monétaire en France. Par rapport à notre modèle précédent, le nouveau modèle semi-structurel a été amélioré dans trois dimensions: les canaux financiers sont plus riches, les anticipations ont à présent un rôle explicite et les simulations convergent à présent vers un chemin de croissance équilibrée. Nous suivons l’approche du modèle FRB/US, dans laquelle les agents forment leurs anticipations de deux façons, à l’aide d’un modèle VAR ou du modèle lui-même, et où les comportements non financiers réagissent avec des coûts d’ajustement polynomiaux. Pour des chocs de politique monétaire standard, FR-BDF témoigne d’une sensibilité plus forte que notre modèle antérieur, en raison de la large influence des anticipations. Nous montrons aussi que, avec des anticipations cohérentes avec le modèle, FR-BDF ne souffre pas du forward guidance puzzle. Enfin, les programmes d’achats d’actifs de l’Eurosystème auraient eu selon FR-BDF des effets notables, avec une transmission passant plus fortement par les taux de change que par les primes de terme.

Mots-clés : modélisation semi-structurelle, anticipations, politique monétaire, guidage prospectif

Les Documents de travail reflètent les idées personnelles de leurs auteurs et n'expriment pas nécessairement

la position de la Banque de France. Ils sont disponibles sur publications.banque-france.fr

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Special acknowledgment

This paper is the outcome of a modeling project with a well-defined governance and manycontributors. As it owes much to many persons within the Banque de France, we would liketo provide here a detailed description of these contributions.

Regarding the governance, first, all choices were continuously validated by M. Lemoine(head of the project) and cross-validated by Jean-François Ouvrard (head of the forecastingunit) as the first user of the model. At the very beginning of the project, its design alsobenefited from inputs from H. Le Bihan (the former head of the forecasting unit). Second,methods and results were also regularly approved by the former director of macroeconomicanalysis and forecasting, P. Sicsic, the current director, G. Levy-Rueff, as well as his deputy,Y. Kalantzis. Third, major choices have also been discussed and validated in an Over-sight Committee, headed by O. Garnier, DG Statistics, Economics and International, andpreviously by the former DG, M.-O. Strauss-Kahn.

Regarding the contributors to the project, the core contributors were the authors of thisdocument, i.e. the members of the modeling team in charge of developing the model: thecurrent members (P. Aldama, H. Turunen and A. Zhutova, headed by M. Lemoine), with theconsultancy of J.-P. Laffargue, as well as earlier members of this team (M. Chahad, P. Clercand A. Lepetit). From the start of the project, this modeling team has been supported by theexcellent assistance of L. Giuliani for the development of databases and the implementationof the accounting framework.

The project also benefited from the expertise of our colleagues on specific topics: C.Thubin on business investment; G. Gaulier and B. Meunier on external trade; H. Camatteon consumption; M. Cochard on labor market; E. Monnet and V. Faubert on householdinvestment; A. Devulder, V. Chouard and M. Lequien on the supply side; C. Malgouyres onwages; J. Matheron, H. Le Bihan and Sarah Mouabbi on the financial block; M. Aouriri andThao Vu on public finances; M. Juillard on simulations under model-consistent expectations;H. Le Damany on equations of demand deflators; Magali Marx and Paul Vertier on inter-actions with HICP forecasts; C. Wolf, S. Périllaud and E. Lor-Lhommet on the insertion ofthe model in the forecasting infrastructure.

Finally, we would like to thank economists from other institutions for useful exchanges:Matteo Ciccarelli, Nikola Bokan, Kai Christoffel and Srecko Zimic for the regular exchangeswe have had during our parallel ECB and Banque de France modeling projects; Flint Bray-ton, Hess Chung and Jean-Philippe Laforte for their explanations about FRB/US and theirfeedback on FR-BDF; José Dorich and his colleagues for their presentations of Bank ofCanada models at the very beginning of the project.

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Contents1 Introduction 4

2 Bird’s-eye view of the model 82.1 Role of expectations in the presence of polynomial adjustment costs . . . . . 82.2 Expectations formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Supply, value added price and labor market . . . . . . . . . . . . . . . . . . 102.4 Demand components and their deflators . . . . . . . . . . . . . . . . . . . . 112.5 Financial block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Accounting framework and public finances . . . . . . . . . . . . . . . . . . . 122.7 Long run of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Expectation formation and the polynomial adjustment costs framework 153.1 Expectations formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Specification of the expectation satellite model . . . . . . . . . . . . . 163.1.2 Estimation of the core of the expectation satellite model . . . . . . . 183.1.3 Estimation results and impulse responses . . . . . . . . . . . . . . . . 21

3.2 Microfoundations for the Polynomial Adjustment Costs framework . . . . . . 233.2.1 Cost functions and rational error-correction equations . . . . . . . . . 243.2.2 Constraint used for ensuring growth neutrality . . . . . . . . . . . . . 25

3.3 Computation of present values . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.1 Present value of expected changes in the target in PAC equations . . 263.3.2 Present values with a constant discount factor . . . . . . . . . . . . . 28

4 Model specification and estimation 294.1 Estimation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Supply block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.1 Theoretical framework of market branches’ supply . . . . . . . . . . . 314.3.2 Equilibrium of the supply of market branches . . . . . . . . . . . . . 344.3.3 Long run output in FR-BDF . . . . . . . . . . . . . . . . . . . . . . . 39

4.4 Value added price of market branches . . . . . . . . . . . . . . . . . . . . . . 414.5 Labor market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5.1 Labor supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.5.2 Labor demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.6 Demand block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.6.1 Household consumption . . . . . . . . . . . . . . . . . . . . . . . . . 534.6.2 Business investment . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.6.3 Household investment . . . . . . . . . . . . . . . . . . . . . . . . . . 664.6.4 External trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.7 Demand deflators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.7.1 Household consumption deflator excluding VAT . . . . . . . . . . . . 754.7.2 Business investment deflator . . . . . . . . . . . . . . . . . . . . . . . 76

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4.7.3 Household investment deflator . . . . . . . . . . . . . . . . . . . . . . 784.7.4 Export deflator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.7.5 Import deflators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.7.6 Government consumption deflator . . . . . . . . . . . . . . . . . . . . 83

4.8 Financial block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.8.1 Short-term interest rate . . . . . . . . . . . . . . . . . . . . . . . . . 844.8.2 Long-term government interest rate . . . . . . . . . . . . . . . . . . . 854.8.3 Private interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.8.4 Exchange rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.8.5 Net property income and net asset positions . . . . . . . . . . . . . . 94

4.9 Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.10 Accounting framework and public finances . . . . . . . . . . . . . . . . . . . 984.11 Model changes for conditional projections within BMPE exercises . . . . . . 100

5 Main model properties under VAR-based expectations 1025.1 Long run convergence in unconditional simulations . . . . . . . . . . . . . . 1025.2 Impulse responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.1 Short-term interest rate shock . . . . . . . . . . . . . . . . . . . . . . 1055.2.2 Term premium shock . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2.3 Foreign demand shock . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.2.4 Government spending shock . . . . . . . . . . . . . . . . . . . . . . . 1085.2.5 Oil price shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2.6 Cost-push shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2.7 Labor efficiency shock . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Monetary policy transmission under different expectation assumptions 1166.1 Alternative version of FR-BDF with model-consistent expectations . . . . . . 116

6.1.1 Equations of present value variables under model-consistent expectations1166.1.2 Simulation methodology . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2 Monetary policy shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.3 Forward guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.3.1 Theoretical background for the forward guidance puzzle . . . . . . . . 1226.3.2 Quantitative comparison . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.4 Asset purchase programmes . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7 Conclusion 131

A Appendix 136A.1 Stability conditions, E-SAT model . . . . . . . . . . . . . . . . . . . . . . . . 136A.2 Implications for the term premium, E-SAT model . . . . . . . . . . . . . . . 136A.3 Additional figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

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1 IntroductionThe model for France of the Banque De France (FR-BDF) is the new semi-structural re-placement of the older medium term macro model Mascotte (Baghli et al., 2004). Due tochallenges posed by the modern economy, certain features of Mascotte had become unsatis-factory: the sensitivity of this model to monetary and financial shocks is very weak, whilesuch shocks are seen as major drivers of the business cycle since the Great Recession. More-over, the traditional macro-econometric approach of Mascotte misses important transmissionchannels of monetary policy related to expectations. As we believe that these missing chan-nels are too important for minor revisions to be satisfactory, the modeling framework shouldbe completely overhauled. This is the starting point of the FR-BDF project.

Before explaining our new approach, we can explain what is the purpose for Banque deFrance of having a large-scale model for France, which contains detailed behavioral equationsas well as a detailed accounting framework (around 400 equations in total). First, for inter-nal purposes, we need a detailed analysis of transmission channels, both for forecasting andcounterfactual exercises. For example, in recent forecasts, it was useful to provide a detaileddecomposition of the effect of fiscal packages related to the yellow vests crisis on households’real disposable income. Second, this level of detail is also important for our participation inthe Eurosystem’s bi-annual broad macroeconomic projections exercises (BMPE), in which allnational central banks of the Eurosystem build the scenario of their own country for a stan-dardized set of variables (around 90 quarterly variables) using a common set of assumptions(Brent price, exchange rates, world demand, etc.).

Participating in the Eurosystem projections also has some other consequences for ourmodeling choices. Although France accounts for roughly 20% of the euro area GDP, a con-scious choice was made to keep the small open economy setup, where French developmentshave no influence on euro area monetary policy and the wider international economy, as wasthe case in Mascotte. The main reason for this choice is that FR-BDF will always be usedin interaction with models from other national central banks and with technical Eurosys-tem assumptions, when conducting Eurosystem operational forecasts. Still, we allow thepossibility of endogenizing some parts of the external environment for some counterfactualexperiments, e.g. the response of exchange rates to monetary policy shocks.

Hence, the modeling project started with several goals in mind. The two most importanttargets to be met were: (i) to generate reasonable and detailed conditional forecasts and (ii)to provide detailed counterfactual scenarios. In particular, FR-BDF should be able to modelthe consequences of monetary policy and financial shocks accurately, and should thus havean explicit role for expectations and a richer financial block. Another desired feature was tohave a smoother convergence to a well-defined balanced-growth path (BGP) in unconditionalsimulations, in order to clarify the distinction between expert judgment and model output.Indeed, judgment should be used more for including external information than for ensuringsuch a convergence.

The overall model structure is strongly inspired by the US model of the Federal ReserveBoard (FRB/US). This type of model combines economically meaningful expectations, agood empirical fit due to the Polynomial Adjustment Costs (PAC) framework, ease of esti-mation due to the separability of model blocks and the possibility of having a well-definedlong-term balanced growth path in general equilibrium. In this approach, we can allow dif-

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ferent types of expectations: expectations based on a small Vector Autoregression (VAR)that summarizes the state of the economy, model-consistent expectations (MCE) and hybridexpectations that would be based on either type of expectation depending on agent types.Still, the main difference between the French and the US economies that has to be reflectedin FR-BDF is clearly the absence of independent monetary policy in the case of France.

In order to achieve the goals described above, we also paid particular attention to thefinancial block of the model. First, the term structure, as well as exchange rates, are en-dogenous, in order to properly analyze the transmission of monetary policy shocks. Second,the structure of interest rates is richer which considerably helps to capture the importantrole of financial shocks.

In comparison with the forecasting models of other major central banks, such as FRB/US,FR-BDF lies halfway between the fully micro-founded DSGEs1 which put more the empha-sis on theoretical consistency and the traditional semi-structural models2 which put moreemphasis on the empirical fit. Compared to other semi-structural models, the FRB/US ap-proach also seeks to achieve a balance between theoretical consistency and data fit, but itrequires additional constraints related to the explicit role of expectations. This FRB/USapproach has also been adopted in some other major central banks: it replaced the DSGEapproach for forecasting exercises at the Bank of Canada; it has also been followed by theECB for the new revision of its semi-structural models involved in the forecasting process.3

Contrary to the Banque de France, other major French institutions have not includedexplicit expectation channels in the recent versions of their forecasting models, but havefocused more on channels that are less important for a central bank than those related tothe transmission of monetary policy. For example, the last version of the Insee-Treasurymodel (Mésange) has been enriched with labor skill segmentation (Bardaji et al., 2017), animportant feature for assessing the impact of labor income tax cuts targeted at low-skilledworkers. Another example is the last version of the OFCE model (e-mod), which has beenenriched with non-linear hysteresis effects in the labor market (Creel et al., 2011) as a wayof capturing the increase in fiscal multipliers during recessions.

The model can also be positioned using conceptual frameworks recently presented inthe academic literature. Blanchard (2018) seeks to classify modern macroeconomic generalequilibrium models into five partially overlapping groups: "core", "foundation", "policy","toy" and "forecasting". Within this classification, even though it is mostly used to forecastthe French economy, FR-BDF is solidly within the "policy" group and, more generally, themodel is intended to be able to provide meaningful answers to counterfactual questions onthe dynamics of the economy.

In this paper, we present the model and its main properties – long-term convergenceand main impulse responses – and we describe the dynamics of the model regarding thetransmission of standard and nonstandard monetary shocks, as this type of simulation yielded

1For examples, see the COMPASS model of the Bank of England, RAMSES of the Riksbank and AINOof the Bank of Finland.

2For examples, see the Makro model of the Bundesbank, BIQM of the Banca d’Italia, MTBE of the Bancode España and DELFI 2.0 of De Nederlandsche Bank.

3These models are respectively ECB-BASE for the euro area version and ECB-MCM for the multi-countryversion. The motive of the ECB for extending their DSGE toolbox and including such semi-structural modelsis explained in detail in the speech of Constâncio (2017), which can be found here.

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particularly unsatisfactory results with our former model. For the analysis of the model’smain properties, we rely on VAR-based expectations, i.e. the type of expectations that weuse in a forecasting context. Then, for experiments related to the analysis of monetarypolicy transmission, we also carry out simulations under model-consistent expectations, asthese expectations might be necessary for capturing the short run impact of announcementsrelated to the persistence of shocks.

As regards the model’s main properties, we start by checking the convergence of uncon-ditional simulations of FR-BDF toward a balanced growth path (BGP). As shown by oursimulations, the output and inflation gaps converge toward zero over a period of around fortyyears. In the short run, the dynamics of FR-BDF can deviate from long run targets due tonominal and real rigidities modeled with PAC. As a result, convergence toward the BGP isachieved once both nominal and real rigidities have vanished. To understand such a slowconvergence, we should keep in mind two key features of the model: in the absence of anindependent monetary policy, these gaps are only closed by very slow price-competitivenessmechanisms; the dynamics of these variables are also influenced by stock variables, i.e. cap-ital services and net financial assets, which have very inertial dynamics.

We then turn to the short run dynamic properties of FR-BDF and present four demandshocks (short rate, term premium, foreign demand and public consumption) and three sup-ply shocks (cost-push on VA price, oil price and labor efficiency). Compared to the impulseresponses of e.g. FRB/US, one striking feature of our impulse responses to positive andtemporary demand shocks is related to the absence of any feedback from monetary policyand nominal exchange rates. The expansionary effect of these shocks on output and infla-tion deteriorates the competitiveness of exports. Net exports and output gap turn negative,leading to disinflation and bringing the real exchange rate back to its baseline level. More-over, the lower inflation that prevails in the medium run pushes real interest rates upwardand amplifies the fall in the output gap. Impulse responses to supply shocks illustrate theimportant role played by the supply block of this model at horizons that could matter formedium-run forecasts. For example, after a shock that gradually raises labor efficiency by1% (by 0.95% after four years), real output increases by 0.6 percentage points over a 4-yearhorizon.

In a last section, we analyze monetary policy transmission in France through three policyexperiments. In our first policy experiment, we assess the impact of a standard monetarypolicy shock, under three different expectations setups – agents may have model-consistentexpectations on just financial variables (hybrid case), on both financial and non-financialvariables (full MCE case) or neither, which corresponds to the VAR case. In all cases,FR-BDF shows a stronger response than our former model and this can be related to thewidespread influence of the short term rate through expectations. There are some differencesin outcomes that depend on how expectations are modeled. When non-financial variablesare modeled in a forward-looking manner (full MCE case), this creates a dampening effect incomparison to the case where they would be backward looking (hybrid case). We can see thisby the fact that the GDP response in the full MCE case is smaller than in the Hybrid case,with peak effects on GDP respectively around 0.1% and 0.3%. If we only model financialvariables in a forward-looking way (hybrid case), we get on the contrary an amplificationeffect in comparison with the VAR-based case: peak effects on GDP are around 0.3% for thehybrid case and around 0.2% for the VAR-based case.

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Second, with the model-consistent expectations version of the model, we can run alter-native simulations in order to assess the macroeconomic impact in France of future policyshocks that would be announced and fully credible, such as forward guidance policies. Withthis version, we examine the impact in France of an announced cut in short-term interestrate for different numbers of quarters. The short run impact of these policies on long-terminterest rates and exchange rates increases with the announced length of the shock. Thesestronger responses of financial variables lead to stronger responses of GDP and inflation.Stronger responses do not mean an explosive amplification: it appears that FR-BDF doesnot suffer from the forward guidance puzzle, i.e. the macroeconomic impact of forward guid-ance does not increase exponentially with the length of the shock, because of the way inwhich the future is discounted in the household consumption behavior. The peak effect onGDP increases almost linearly with the duration of the shock: after a shock announced tolast 8 quarters, the peak effect on GDP is amplified by around 70% compared to the oneobtained after a 1-quarter shock, which is slightly more than the double of the amplificationobtained after a 4-quarter shock (around 30%).4

The third experiment we consider deals with how the asset purchase programmes (APP)conducted by the Eurosystem have affected the French economy. These programmes pro-vide monetary accommodation by transferring liquidity directly to market participants inexchange for various assets. The empirical literature has identified effects on the term pre-mium, but also on the exchange rate. We use these estimates to study the response of theFrench economy. We construct a counterfactual outcome for the French economy by remov-ing the effects of APP on the term premium and exchange rates using estimates of shocksto these quantities stemming from APP. The simulations are conducted under VAR-basedand model-consistent expectations, assuming that the French economy is otherwise discon-nected from the rest of the euro area. Our main results show that APP had notable realand nominal effects on the French economy. Overall it would seem that APP is a shock thatboosts exports and investment through a depreciation of the exchange rate and a fall in longinterest rates. On the nominal side, it first results in imported inflation and then in domesticinflation due to an increase in factor costs. The main difference in outcomes due to the wayin which expectations are modeled is in the response of inflation, which reacts much morestrongly under model-consistent expectations.

The rest of the note is structured as follows. The next section provides a bird’s-eye viewof the overall features of FR-BDF. Section 3 discusses both expectations formation and thePAC framework. A detailed description of the different blocks of the model is presented insection 4. Section 5 is devoted to the model’s main properties, while section 6 focuses on thetransmission of standard and nonstandard monetary policy shocks. Section 7 concludes.

4In recent applications of DSGE models to the analysis of the consequences of forward guidance, a methodof mitigating the puzzle consists of applying discounted Euler equations. See for example Nakata et al. (2019)and some examples that we provide in section 6.3.

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2 Bird’s-eye view of the modelThis section gives an overview of the main blocks of FR-BDF. We describe the model asit would be applied for most types of policy analysis and unconditional simulations. Forforecasting (e.g. Eurosystem macroeconomic projections) various model components wouldbe exogenized to comply with the assumptions of those exercises (see last subsection).

2.1 Role of expectations in the presence of polynomial adjustmentcosts

For non-financial behavioral equations, the key role played by expectations is a consequenceof applying the PAC framework, where, starting from an initial condition that differs fromtheir long-term targets or after a shock that affects these long-term targets, agents adjusttheir decision variables toward the expected path of these long-term targets.

The agents’ long-term targets – conceptually independent of the rest of the model –are determined by economic theory in a flexible economy, which is close to a neoclassicaleconomy augmented with monopolistic competition. For example, firms’ employment target,constructed from the production function and the associated first order condition, dependson aggregate demand and real labor costs. In some cases, some targets can directly dependon expected variables: for example, the consumption target depends on permanent income,defined as the expected average of future income flows.5

In the short run, economic agents seek to set the trajectory of a variable that they careabout (e.g. employment) so as to minimize deviations from the long-term target (as describedabove) under PAC. Rearranging the first order condition of this problem leads to what arecalled the short run equations of FR-BDF, which describe the dynamics of the model’s mainvariables given the dynamics of the target. A crucial feature of these short run equationsis the presence of expectations regarding the target – variables that describe agents’ beliefsregarding the future state of the economy. These expectations arise out of the adjustmentcosts – given their expected future state of the economy, agents know that their choices inthe future are quite possibly different from today’s choices; these constraints imply that itis prudent to adjust today’s choices somewhat towards their future choices. For example,for a recessionary shock that would be expected to be temporary, the expected employmenttarget would not fall as much and firms would cut fewer jobs (labor hoarding) than theywould for shocks expected to be persistent.

In the financial block, adjustment costs play no role but expectations still appear throughno-arbitrage conditions, namely the term structure of interest rates and the uncovered in-terest rate parity condition used for modeling exchange rates.

Figure 2.1.1 presents a simplified diagram of our model where variables directly affectedby expectations appear in red, demonstrating how expectations play a widespread role inthe whole model, i.e. in most of the non-financial and financial blocks.

5One exception is the investment target which does not necessarily set capital services equal to theirflexible level in the short run (see subsection 4.3).

8

EXTERNAL TRADE BLOCK: - Imports: internal demand

and price competitiveness - Exports: external demand

and price competitiveness

SUPPLY BLOCK: - Long-run output determined by

production function - Labor demand, employment

and unemployment - Capital accumulation

DEMAND BLOCK: - Consumption: permanent

income and bank lending rate; presence of HtM agents

- Business investment: cost of capital and aggregate demand

- Household investment: cost of capital and permanent income

NOMINAL BLOCK: - Gross wage: expected unemployment (Phillips) - Cost of capital: WACC, expected inflation - Domestic price: cost of production factors - Demand deflators: domestic and import prices

FINANCIAL BLOCK: - Term structure of interest rates - Exch. rate: uncov. int. rate parity condition - Credit to households and firms - Net financial assets of each agent

Expectations VAR-based or

model consistent

In red, variables directly affected by expectations

PUBLIC FINANCE BLOCK: - In simulation: receipts driven

by effective tax rates & tax bases / spending driven by long-run output

Figure 2.1.1: Simplified scheme of FR-BDF showing the widespread role of expectations

2.2 Expectations formation

In FR-BDF there are three types of expectations formation: (i) backward-looking expecta-tions based on a satellite model called "E-SAT" (also referred to as VAR-based expectations),(ii) model-consistent expectations (MCE), (iii) a combination of these two (hybrid expec-tations). In the first case, agents base their expectations on a smaller and simpler modelwhich is supposed to capture the main features of the economy (represented by the FR-BDFmodel). In the MCE case, agents are forward-looking and their predictions coincide withthe FR-BDF forecast: they behave in a forward-looking manner with perfect foresight, i.e.they have a perfect knowledge of the future dynamics of the models’ exogenous variables.Moreover, we are able to consider hybrid expectations. For example, Bernanke et al. (2019)performed simulations of FRB/US with MC expectations for financial agents, assumed to bewell-informed, and VAR-based expectations for non-financial agents, assumed to have morelimited information, in order to study the consequences of several monetary strategies.

E-SAT is a structural VAR model estimated with Bayesian methods. Its core is formedby IS and Phillips curves for the euro area and France and a Taylor rule for the euro area. Toensure consistency between E-SAT and the full model, we also make a strong simplificationthat France does not influence the euro area. The French IS curve relates the French output

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gap to the interest rate set by the governing council of the ECB. In order to take into accountthe sensitivity of economic developments in France to those in the euro area, this IS curvealso relates the French output gap to the euro area output gap in a reduced form.

The model is completed by equations describing the behavior of 3 anchor variables. Inthe long run, inflation (both in France and in the euro area) and the interest rate are assumedto converge toward these long-term anchors. These long run anchors are measured eitherusing survey-based or market-based expectations.

E-SAT can also be applied to compute expectations for variables in FR-BDF that are notwithin the previously described core. In this case, E-SAT has to be augmented with auxiliaryequations that describe how the variable(s) of interest are related to the core variables.6 Asan example, business investment is dependent on an expectation concerning the deviation ofmarket value added from its trend. The auxiliary equation relates the deviation of currentvalue added to its lag and the output gap, which is in the E-SAT core.

In practical terms, contrary to what is usually done in the DSGE literature, equationsare solved forward and agents build their expectations about infinite sums of variables ofinterest from the next period to the long run. In most cases, these sums are discounted. Ineach short run equation, the discounted sum of expected changes of the long-term target iscomputed using the PAC framework. In such cases, subjective discount factors of agents areadjusted to take into account the presence of adjustment costs. For example, when adjustingemployment is very costly, firms will be more sensitive to future economic outlook relativeto the current one for deciding the current level of their employment.

Finally, in the VAR-based case expectations appear as policy functions obtained by in-verting the VAR model. In the MCE case expectations enter the model as they are definedusing a forward-looking recursive form, i.e. with a finite number of leads of the variableitself.

2.3 Supply, value added price and labor market

We define long run output as the level of production that could be reached with full-utilizationof the capital stock and the exogenous long run level of the unemployment rate. It isdetermined with a Constant Elasticity of Substitution (CES) production function of firms,which we proxy with the value added of market branches. This production function usescapital and labor as inputs with exogenously growing labor-augmenting technical progress.

In the short run, output is determined by demand components, through the clearingcondition of the market of domestic goods and services. Capital services are determined bya standard accumulation equation that depends on investment which is determined in thelong run by firms’ first order condition with respect to capital (described in Section 4.3).

The value added price set by firms is modeled with a PAC equation. The long runprice target is determined by a price frontier consistent with the CES production function,labor-demand condition and a markup stemming from monopolistic competition.

Labor demand is determined in the long run by firms’ first order conditions with respectto labor, while polynomial adjustment costs smooth short run dynamics consistently with

6Note that the auxiliary equations are estimated separately and the core E-SAT coefficients stay un-changed.

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labor hoarding observed on French data. Labor supply is determined by a wage Phillips curveà la Gali (Gali et al., 2011) that mainly relates wage growth to the expected unemploymentgap. In the long run, this curve is vertical and the unemployment rate is set equal to itsexogenous long run level.

2.4 Demand components and their deflators

The target of the main demand component, household consumption, is determined by per-manent income and by an interest rate gap. Permanent income is constructed using a highdiscount rate that arises from a combination of household risk aversion and income uncer-tainty. Short run dynamics are determined via an estimated PAC equation that is extendedwith a term representing the current output gap, in order to take into account the sensitivityof hand-to-mouth consumers to current income.

Households formulate their investment target based on the same permanent income termas for consumption. Their decision also depends on the user cost of household capital andon relative prices of new and existing housing. This price sensitivity reflects their higherincentive to invest when the price of old housing is at a high level relative to the price ofnew housing.

The target for business investment is obtained through a steady-state investment-outputratio derived itself from first order conditions of the firms’ optimization problem. It relatesdesired investment to output and to the user cost of capital. In the short run, following thestandard Tobin’s Q theory of investment, the presence of adjustment costs, polynomial inour framework, creates some stickiness of investment with respect to its target dynamics.Moreover, this PAC equation is extended with an ad hoc term – current value added growth– which improves the fit of the equation, one possible reason for this being the role ofliquidity-constrained firms in investment dynamics.

As regards external trade, the target of real imports (both energy and non-energy) ismainly driven by the real exchange rate and an import intensity-adjusted measure of aggre-gate demand for imports, which in turn is determined as a weighted sum of the other demandcomponents, including exports, with weights equal to their import content shares. For realexports, the target is mainly driven by world demand and the real exchange rate. The shortrun dynamics of real imports and exports are governed by ECM equations instead of fol-lowing the PAC framework. These equations relate the growth rates imports and exportsto the long run real growth rate of the economy and to either internal or world demand,along with an error correction component, i.e. the difference between the lagged level of theappropriate volume and its target.

Demand deflators are modeled with simple error-correction equations, with targets de-fined as weighted averages of domestic and import prices, the domestic price being the valueadded (VA) price of market branches and the import price being mainly determined by theprice of competitors and the oil price.

2.5 Financial block

The short rate (3 month Euribor) is the main interest rate of the model – variation in thisrate causes variation in the long-term government bond rate, which causes variation in the

11

private interest rates. In simulation, it is based on either a simple AR(1) – with a long runanchor given by the historical mean – in order to abstract from issues related to the euroarea, or on a Taylor rule that reacts to euro area inflation and output gap.

The 10-year government bond rate has an important role as the rate on which otherlong-term rates are based (long-term bank lending rates, as well as the corporate bond rateand cost of equity). In simulation, it is determined with a term structure equation – thelong rate is simply the sum of an expectation component and a time-varying term premium.

Exchange rates are determined via uncovered interest rate parity (UIP). Two exchangerates are computed: the rate between the euro and the US dollar and the rate betweenthe euro and a basket of currencies used by France’s other trading partners. These ratesare then applied to determine the price of oil in dollars and the price levels of the France’scompetitors.

Finally, Figure 2.5.1 provides a simplified diagram of the FR-BDF financial block. First,the short-term rate influences all expectations, even in the VAR-based case. Second, arrowsshow the transmission channels, in particular the direct effect of the short-term rate onthe long-term rate through the term structure and on exchange rates through the UIPcondition. Third, the effect of interest rates on business investment transmits through thethree components of the weighted average cost of capital (WACC), i.e. the bank lending rate,the bond rate and the cost of equity. Interest rates also influence households’ consumptionand investment choices through their bank lending rate.

Variables Effect in FR-BDF

Short run interest rate All expectations

Long run interest rate All private interest rates

Bank lending rate households Demand of households

Return rate of financial assets Net financial income

Bank lending rate firms Business investment

Bond rate of firms Business investment

Cost of Equity Business investment

Exchange rate External trade

Figure 2.5.1: Simplified diagram of the financial block of FR-BDF

2.6 Accounting framework and public finances

The accounting framework of FR-BDF mimics the quarterly national accounts with greatdetail and is structured around two decompositions:

• A branch decomposition (market/non-market) of the value added account and of labormarket variables and a product decomposition (energy/non-energy) in the specific caseof imports;

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• Supply and use accounts in value and volume, as well as sector accounts in value forfive institutional sectors (firms, households, government, non-profit organizations andthe rest of the world).

Consistency between these axes is ensured through bridge equations, i.e. production andlabor market variables of sector accounts (value added and wages) are extrapolated withsimple equations which relate them to corresponding variables from the branch decompo-sition. For example, the nominal value added of firms is extrapolated with a proportionalfactor with respect to the one of market branches.

The government block is particularly detailed and designed in such a way as to allowinteractions with the public finance model of the Banque de France. In particular, for manyvariables of this block two modes were created: a forecasting mode, where many nominalvariables are exogenized and come directly from the public finance model; a simulationmode, where these variables are endogenous and depend on exogenous effective tax rates orexogenous ratios. More precisely, in the simulation mode, we adopt the following commonprinciples on the receipt and spending sides:

• On the receipt side, each receipt is determined by an exogenous effective tax rate andan endogenous tax basis;

• On the spending side, some spending variables are directly related to macroeconomicaggregates with effective rates. For example, unemployment benefits are directly re-lated to unemployment and wage per capita. For other spending variables (excludingsocial transfers as explained below), e.g. intermediate consumption or governmentinvestment, the ratio of their volume relative to long run output is assumed to beexogenous.

In simulation, we endogenize a fiscal rule, which assumes that an instrument – social transfersin our baseline case – will be used by the government to ensure the convergence of its netasset ratio toward its long run target.

2.7 Long run of the model

In the long run, the variables of the model follow a balanced growth path, where, on the onehand, output growth is determined by labor efficiency and demography, and, on the otherhand, inflation is determined by the ECB inflation target. When the price frontier and firstorder conditions with respect to labor and capital are met, the level of output endogenouslyconverges toward the long run output (defined in the subsection describing the supply block).Contrary to closed-economy models, the real interest rate here is exogenous in the long run,determined by the exogenous nominal interest rate and by the ECB inflation target. Settingthe growth rate of world demand and the inflation rates of competing countries equal torates at the domestic level for ensuring a balanced growth path, the convergence of demandtoward the long run output determines, in our small open economy framework, the long runequilibrium of the real exchange rate, a key driver of external trade (as shown in section 4.6).As competitors’ prices are exogenous, as well as the nominal exchange rate in the long run,the level of the real exchange rate determines the level of the domestic value added price inthe long run.

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2.8 Estimation

The estimation of equations requires specifying the way in which agents form their expecta-tions. As in FRB/US, we choose here to estimate our model under VAR-based expectations,in order to be able to estimate the model block by block. Although this might create simul-taneity biases, this approach seems more suited to such a large-scale model for two reasons.First, this approach has the advantage that potential misspecifications of some blocks willnot spoil the whole estimation of the model, as would be the case with joint estimation.Second, this makes the estimation more flexible: a joint estimation would be very difficultand the cost of conducting a fully fledged joint re-estimation each time a single block isrevised would be too large. Our backward-looking expectations stem from our structuralBayesian VAR. This VAR is used to generate the various expectation terms of the model.In particular, we estimate short run PAC equations with an iterative OLS procedure à laFRB/US which iterates two steps: the estimation of short run coefficients conditional onexpectation terms; the computation of expectation terms with the VAR and conditional onthese coefficients. We provide more details about estimation in Section 4.1.

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3 Expectation formation and the polynomial adjustmentcosts framework

In this section, we explain the way in which agents form their expectations and the reasonswhy these expectations play a role in the model through the present value of future changes intargets in most short run equations, due to the presence of polynomial adjustment costs. Thissecond topic is split into two subsections devoted to the micro-foundations of the polynomialadjustment costs framework, on the one hand, and the computation of present values, onthe other.7

3.1 Expectations formation

Expectations are at the core of modern macroeconomic models as economic decisions (con-sumption, investment, etc) depend on future conditions. The better expectations are incor-porated the more reliable is the model. Expectations are supposed to dampen or amplifythe impact of certain shocks: e.g. to dampen the employment response in the case of atemporary negative demand shock or to amplify the response of financial variables when amonetary policy shock is announced to last longer than expected. The presence of expecta-tions is one of the main advantages of FR-BDF with respect to the former forecasting modelof the Banque de France (Mascotte). This feature of FR-BDF enables the study of questionsconnected to monetary policy or macro-financial issues.

There are two types of expectations formation in FR-BDF. The first type, our base-line case described in this section, assumes that agents have limited knowledge about themodel. They form their expectations based on a simpler model containing much fewer vari-ables, which we refer to as the Expectation SATellite model (E-SAT). These expectationsare backward-looking and referred to as VAR-based expectations because in this case ex-pectations are identical to the forecast of a constrained VAR model. The second type ofexpectations formation, described in section 6.1, is forward-looking: agents have full knowl-edge about the model and they can adjust their decisions to information about the futurethe moment they receive it. These expectations are referred to as "model-consistent".

In some practical experiments (see section 6) we use a hybrid framework in which financialagents form their expectations using the second method while others have limited informationto take their decisions and base their expectations on a simple VAR model. We believe thatthis is an interesting exercise as there is still no consensus about how agents form theirexpectations. For example, one may argue that agents making financial decisions wouldprefer to take into account any available information about the economy even if it is costly,unlike a decision on short run consumption.

FR-BDF contains 11 variables for which agents form expectations. Some of them appearin the short run behavioral equations of non-financial agents, due to the presence of adjust-ment costs. This concerns the short run equations of household consumption and investment,corporate consumption, employment, wage and value added price. Other expectation vari-ables (permanent income and inflation) appear directly in the long run equations of the

7Please note that this section treats only backward-looking expectations, i.e. based on a small VAR.Model-consistent expectations are discussed in 6.1.

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desired targets (household consumption and corporate investment respectively). Finally, ex-pectation terms are found in asset pricing equations which are derived from no-arbitrageconditions as well as in simple discounting schemes: domestic and foreign short run interestrates.8

In what follows we describe the FR-BDF VAR model that agents use to form theirexpectations when they do not have model-consistent expectations. For the details abouthow we compute the present values of expected variables, we invite the reader to consultsection 3.3.

3.1.1 Specification of the expectation satellite model

The expectation satellite model E-SAT consists of a core, with eight equations and an aug-mented part which varies depending on blocks to be estimated.

Semi-structural equations and VAR representation The core of E-SAT is based ona small model with two blocks: France and the euro area.9 The model consists of the IS andPhillips curves for each block and is completed with a Taylor rule which sets the interestrate as a function of EA inflation and its output gap. This model is a structural VAR modelin the spirit of Rudebusch & Svensson (1999), where we add shifting endpoints of inflationand the interest rate. As shown in Kozicki & Tinsley (2001), taking into account shifts inendpoints supports a more substantial term structure role for short rate expectations. Thecore E-SAT model has eight equations:

(1− λqL)yt = −σq(it−1 − πQ,t−1 − ıt−1 + πt−1) + δqyea,t + εq,t

(1− λπL)(πQ,t − πt) = κπyt−1 + επ,t

(1− λiL) (it − ıt) = (1− λi)(αi(πea,t−1 − πt−1) + βiyea,t−1) + εi,t

(1− λq,eaL)yea,t = −σq,ea(it−1 − πea,t−1 − ıt−1 + πea,t−1) + εq,eat

(1− λπ,eaL)(πea,t − πea,t) = κπ,eayea,t−1 + επ,eat

(1− λıL)(ıt − ı) = εı,t

(1− λπL)(πt − π) = επ,t

(1− λ ¯π,eaL)(πea,t − πea) = επ,ea,t

where L stands for the lag operator. The IS curve of each sector relates the output gap tothe real interest rate gap (with respect to the shifting endpoint ıt of the interest rate). Wealso take into account a co-movement between the French output gap yt and that of theeuro area yea,t through a global demand (trade) channel and not only through the ECB’smonetary policy. The Phillips curve relates domestic inflation πQ,t to the inflation shiftingendpoint πQ and the output gap. The inertial Taylor rule relates the interest rate it to itslag, euro area inflation πea,t and the euro area output gap yea,t.

8Here we have listed 11 expectation variables. In order to obtain 11, we take into account that we have twotypes of present values of domestic short run interest rate: a discounted present value and a non-discountedone.

9We do not consider any other foreign variables since: (i) we are limited in the number of variables withinESAT, and (ii) EA variables may be viewed as an approximation for the rest of the world.

16

We have also considered an alternative specification of the ESAT model, where the mainstabilization mechanism is ensured through a real exchange rate adjustment: a Taylor ruleis replaced with an AR(1) process of the real interest rate, and an IS curve is augmentedby an additional term log(pt) − log(pea,t) + log(et), where et is an exchange rate for euroarea countries and is supposed to vary around 1. This model is perfectly consistent with theassumptions of FR-BDF, but the convergence of that specification is much longer than inthe baseline.10

Forming the A and B matrices for applications We can represent the E-SAT modelas a structural VAR of a vector Zt of eight variables and an intercept: AZt = BZt−1 + εt.The A and B matrices have dimension 9× 9. We can deduce from this structural form thefollowing reduced form:

Zt = HZt−1 + ηt (1)

with H = A−1B, ηt = A−1εt. Finally, the reduced form directly delivers the following i-stepahead forecast: Ze

t+i = H iZt.Given the following order of variables in vector Z = [1, y, i, πQ, yea, πea, ı, π, πea], the

expressions of A and B are easily deduced from the equations of the model:

A =

1 0 0 0 0 0 0 0 00 1 0 0 −δq 0 0 0 00 0 1 0 0 0 −1 0 00 0 0 1 0 0 0 −1 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 −10 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1

B =

1 0 0 0 0 0 0 0 00 λq −σq σq 0 0 σq −σq 00 0 λi 0 βi(1− λi) αi(1− λi) −λi 0 −αi(1− λi)0 κπ 0 λπ 0 0 0 −λπ 00 0 −σq,ea 0 λq,ea σq,ea σq,ea 0 −σq,ea0 0 0 0 κπ,ea λπ,ea 0 0 −λπ,ea

(1− λı)i 0 0 0 0 0 λi 0 0(1− λπ)π 0 0 0 0 0 0 λπ 0

(1− λ ¯π,ea)πea 0 0 0 0 0 0 0 λ ¯πea

These eight equations which form the core of the ESAT model are not always sufficient

to describe the formation of agents’ expectations, which may bear on other variables, absentfrom E-SAT. This will require adding auxiliary equations into the system. Assume forinstance that the model includes expectations on the future values of a target variable n∗t . Inthis case agents have to foresee the changes in the trend of n∗t , denoted n∗t , and the changesin its cyclical component n∗t . We then add to E-SAT two auxiliary equations, which may be,

10Details are available upon request.

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for instance

(1− λn∗L) n∗t = an∗yt−1 + bn∗ (it−1 − ıt−1) + cn∗ (πQ,t−1 − πQ,t−1) + εn∗,t∆n∗t = ∆n∗t−1 + εn∗,t

With these two added variables vector Zt becomes

Zt = (1, yt, it, πQ,t, Qea,t, πea,t, ıt, πQ,t, πea, n∗t ,∆n

∗t )′

Matrices A and B are now 11x11. The added rows of A are:

A10 =[

0 0 0 0 0 0 0 0 0 1 0]

A11 =[

0 0 0 0 0 0 0 0 0 0 1]

Those of B are

B10 =[

0 an∗ bn∗ cn∗ 0 0 −bn∗ −cn∗ 0 λn∗ 0]

B11 =[

0 0 0 0 0 0 0 0 0 0 1]

Finally, the additional columns - above rows 10 and 11 - of A and B are simply filled withzeros, as the core variables are assumed to be unaffected by the auxiliary variables.

3.1.2 Estimation of the core of the expectation satellite model

Methodology We estimate the core system (the first five equations of the system writtenin deviation from long run anchors, in subsection 3.1.1) using Bayesian techniques and de-trended data. The choice of applying Bayesian estimation instead of maximum likelihood isdriven by two factors: a relatively small sample and potential misspecification of the model.

The sample that we use to estimate E-SAT is not very large and the data might thereforenot contain sufficient information to identify (with precision) the parameters. When thereis a lack of identification, the likelihood function is flat in certain directions. In this casethe Bayesian approach is more efficient and less sensitive to sample size, since priors canbe used to introduce "curvature" into the objective function. From a numerical perspectivemaximizing the posterior is "easier" than maximizing the likelihood function. The priorinformation may also be helpful to discriminate between hypotheses.

Sample size is not the only problem that we have to deal with. It is difficult to matcheuro area inflation with a simple Phillips curve.11 This is due to the fact that all E-SATvariables enter the policy functions of expected terms, and more complicated policy functionsare harder to interpret during analysis. When we estimate E-SAT with maximum likelihood(ML), depending on the initial value of the estimation, the αi weight of inflation in theTaylor rule is either negative, or below one and not significant.12 Due to the stylized andoften misspecified nature of DSGE (VAR) models, the likelihood is known to often peak inregions of the parameter space that are contradictory with common observations, leadingto the "dilemma of absurd parameter estimates" (Adjemian & Pelgrin (2008), Schorfheide

11For example in the last versions of FRB/US, in order to recover parameters of the inflation equation,wage and price Phillips curves are estimated jointly.

12We also tried to reduce the E-SAT to a model of the euro area economy of three equations and to excludethe zero lower bound region, but the results remained unchanged.

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(2011)). αi was not the only parameter that we had difficulties estimating with the MLmethod. The parameters σq, σq,ea and βi also suffered from the misspecification.

The priors are specified in Table 3.1.1. We are aware of the criticism levelled at theBayesian approach saying that by "playing" with priors we may impose a desirable outcome.In order to address this concern, we ensured that our results were robust with a differentspecification for our priors, which we always chose so as to stay within the boundaries ofconventional wisdom.

Some of the parameters of the equations of the long run anchors are calibrated and othersare estimated, see table 3.1.3. The steady state of annualized inflation in France and theeuro area are set to the ECB target: 1.9%. The persistence λı of the long run anchor ofthe interest rate has strong implications for the term premium. For the reasons explained insection A.2 we calibrate this parameter. Other parameters are estimated using OLS equationby equation on a pre-crisis sample: 1999Q1 - 2008Q4.

Figure 3.1.1: Time series used for the estimation of E-SAT

2000 2005 2010 2015

0

2

4

Annual interest rate LR expectations

2000 2005 2010 2015

-1

0

1

2

3

Annual VA french inflation LR expectations

2000 2005 2010 20150

1

2

3

Annual GDP EA deflator LR expectations

2000 2005 2010 2015

-1

0

1

Quarterly french output gap

2000 2005 2010 2015

-2

0

2

4

Quarterly EA output gap

Data The estimation period is 1999Q1-2017Q4 and time series are shown in Figure 3.1.1.Below we provide the description of data construction and its sources.

• The French output gap is calculated as a difference between actual and long run valueadded.13

13To estimate E-SAT, we use a vintage of the output gap estimate which is slightly different from the

19

Table 3.1.1: Priors and estimation results

Prior Posterior distributiondistribution Mode St Dev Mean 10% 90%

σεq Inv. Gamma (0.1,2) 0.33 0.03 0.34 0.29 0.39σεpi Inv. Gamma (0.1,2) 0.26 0.02 0.26 0.23 0.30σεπ,ea Inv. Gamma (0.1,2) 0.18 0.01 0.19 0.16 0.21σεi Inv. Gamma (0.1,2) 0.09 0.01 0.10 0.08 0.11σεq,ea Inv. Gamma (0.1,2) 0.56 0.05 0.58 0.50 0.66δ Normal(0.2,0.2) 0.08 0.03 0.08 0.03 0.12λq Beta(0.5,0.2) 0.73 0.06 0.73 0.62 0.84λq,ea Beta(0.5,0.2) 0.94 0.03 0.93 0.89 0.97σq Normal(0.6,0.15) 0.27 0.10 0.28 0.11 0.45σq,ea Normal(0.6,0.15) 0.54 0.13 0.54 0.33 0.76λπ Beta(0.5,0.2) 0.58 0.09 0.58 0.43 0.72λπ,ea Beta(0.5,0.2) 0.34 0.10 0.35 0.19 0.51κπ Gamma(0.5,0.2) 0.07 0.03 0.08 0.02 0.13κπ,ea Gamma(0.5,0.2) 0.04 0.01 0.04 0,.2 0.05λi Beta(0.3,0.15) 0.93 0.03 0.92 0.88 0.97α Normal(1.5,0.25) 1.22 0.28 1.19 0.71 1.66β Gamma(0.2,0.1) 0.07 0.04 0.09 0.02 0.16

• The euro area output gap and euroa area inflation are taken from the Eurosystem’sforecasting exercise from March 2018. The euro area inflation is computed with theGDP deflator.

• The French inflation rate is measured as the quarterly growth rate of the value addeddeflator.

• The interest rate is the 3-month Euribor, interpolated with Eonia between 1995Q2 and1999Q4.

• Long run expectations of both inflation rates come from the long run professionalconsensus forecast surveyed by the private company "Consensus Economics". Thelong run horizon is here an average of horizons going from 5 to 10 years. The long runexpectation of euro area inflation is retropolated before 2002 with a weighted averageof five countries: France, Italy, Germany, the Netherlands and Spain.

one used in the rest of the model. As detailed in section 4.3, long run value added of market branches iscalculated using a measure of capital services for which we need wealth accounts. Because wealth accountsin base 2014 were not available at the time of the estimation of E-SAT, we used wealth accounts in base2010 after having re-based the deflators of capital and investment sub-components in 2014. Still, we checkedthat the last version of the output gap would not have substantially changed these results if we had used thewealth accounts in base 2014.

20

Table 3.1.2: Posterior mean variance decomposition (in %)

σεi σεpi σεq σεπ,ea σεq,eayt 39.7 6.26 31.92 1.72 20.4(πQ,t − πt) 11.25 77.76 5.48 0.5 5.02(it − ıt) 67.35 0 0 7.32 25.33yea,t 38.52 0 0 2.84 58.64(πea,t − πea,t) 11.83 0 0 73.48 14.69

• The long run expectation of the interest rate comes from 5-year futures of the 3-monthEuribor.

The long run expectation of the interest rate is shifted by a constant to make it sharethe same average as the 3-month Euribor.14

Table 3.1.3: Parameters of the long run expectations

Estimated Calibrated

ı 3.68 λi 0.985λπea 0.93 π 1.9λπ 0.93 πea 1.9

ı, π and πea are annual rates in p.p.

3.1.3 Estimation results and impulse responses

We use the mean of the posterior distribution as the point estimate of parameters in FR-BDF.All the parameters (except σq,ea) seem to be well identified and the posterior distributiondiffers from our priors, see Figure A.1 in section A.3.15 The values of parameters confirmconventional wisdom. They also meet the stability conditions of the core block of E-SATdescribed in appendix A.1. As regards the disturbances, as shown by the posterior meanvariance decomposition (Table 3.1.2), the interest rate shock seems to play an important rolein explaining the dynamics of the French and the euro area output gap.

In this subsection, we look at the response of the output gap and inflation to shocksto two variables: the interest rate and the euro area output gap. For computing impulseresponses, we also use the mean of the posterior distribution as a central tendency.16 We

14In the current version we decided not to shift the long run expectations of French and euro area inflationrates, i.e. to assume that the long run expectation for value added inflation would be the same as for HICPinflation.

15Identification of parameters is a vague notion. Not all of the parameters are well identified "globally",irrespective of the estimation approach. It seems that without priors, the posterior distribution is ratherflat. When we used the ML method we encountered a problem estimating three parameters: αi, δq and σq.Their values neither have a conventional signs nor are they significant.

16We also checked the impulse response functions (IRFs) computed using the mode of the posterior dis-tribution and the results were unchanged.

21

find that in both cases the impulse responses have humped shapes, with expected signs andresponses close to zero after around 20 years.

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

25 50 75 100

Interest rate (annualized)

-.4

-.3

-.2

-.1

.0

.1

-.4

-.3

-.2

-.1

.0

.1

25 50 75 100

French Output gap

-.25

-.20

-.15

-.10

-.05

.00

.05

-.25

-.20

-.15

-.10

-.05

.00

.05

25 50 75 100

French VA inflation (annualized)

-.8

-.6

-.4

-.2

.0

.2

-.8

-.6

-.4

-.2

.0

.2

25 50 75 100

EA Output gap

-.20

-.16

-.12

-.08

-.04

.00

.04

-.20

-.16

-.12

-.08

-.04

.00

.04

25 50 75 100

EA VA inflation (annualized)

Figure 3.1.2: Impulse responses for E-SAT of French and euro area variables to an interestrate shock

Response to an interest rate shock First, we look at the properties of key impulseresponses of the satellite model to a 0.25pp interest rate hike (i.e. 1pp for an annualizedrate), with a historical persistence of 0.93 (Figure 3.1.2). In the short run, this hike has adirect negative impact on both output gaps and, hence, on inflation in the euro area as wellas in France. The fall in inflation amplifies the rise in the real rate and, hence, the fall indemand due to the Taylor rule inertia. This generates a hump after around three years, witha trough in the output gap at −0.34pp and of value added inflation (annualized) at −0.24pp:obviously, the Phillips curve slope is rather steep: κπ/(1−λπ) ·4 = 0.076/(1−0.43) ·4 = 0.71in annual terms. This estimate is higher than 0.5 reported in Chatelais et al. (2015). As theinterest rate shock is temporary, the output gap and inflation return to their steady stateafter this hump.

The sensitivity of the euro area output gap with respect to the change in the real interestrate seems to be much stronger than in the case of France. This is consistent with theestimated σq,ea which is almost twice as high as σq and also higher than the persistence ofeuro area output gap, see Table 3.1.1.

The reaction of the euro area inflation is much smaller than that in France. The impliedPhillips curve slope is 0.22 in annual terms.

If we compare these results with those of the workhorse VAR study of Peersman & Smets(2001) based on euro area data, we find that our results are not far from theirs. If we rescaletheir shock to 100 bp on the annualized interest rate, the impact after 12 quarters is −0.5%for output and −0.3% for consumer prices. Regarding output, our response is slightly weaker.As for consumer prices, to facilitate the comparison, we look at the cumulative response of

22

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

25 50 75 100

EA Output gap

-.10

-.05

.00

.05

.10

.15

.20

.25

-.10

-.05

.00

.05

.10

.15

.20

.25

25 50 75 100

French Output gap

-.050

-.025

.000

.025

.050

.075

.100

.125

.150

-.050

-.025

.000

.025

.050

.075

.100

.125

.150

25 50 75 100

French VA inflation (annualized)

-.04

.00

.04

.08

.12

.16

.20

.24

.28

-.04

.00

.04

.08

.12

.16

.20

.24

.28

25 50 75 100

Interest rate (annualized)

Figure 3.1.3: Impulse responses for E-SAT of French and euro area variables to a shock toeuro area output gap

inflation and obtain an impact on the value added price level of −0.5pp after 12 quarters,which is stronger than theirs.

Response to a shock to the euro area output gap Here, the dynamics of the variablesare triggered through two channels: the trade channel and the monetary policy channel, seeFigure 3.1.3. Due to the trade channel included in the French IS curve (δyea), we obtain apositive spillover effect of the euro area shocks to France.

3.2 Microfoundations for the Polynomial Adjustment Costs frame-work

The foundation of the Polynomial Adjustment Costs framework (PAC) – which we use tomodel most of our non-financial behavioral equations – is an extension of ideas familiar tomany from the Dynamic Stochastic General Equilibrium (DSGE) literature. The frameworkcan be characterized as a generalization of the modeling devices used in e.g. Rotembergpricing (Rotemberg, 1982), where firms seek to minimize the deviation between the targetprice of the good they are selling and the actual price they set today under quadratic costsof adjustment. The PAC framework takes this idea further by making the cost functionpolynomial: not only is it quadratically costly to deviate from the target, but also the mlatest differences are penalized in the PAC cost function. Furthermore, the PAC frameworkalso implies an error correction equation – augmented with an expected present value offuture changes in the target – that can be obtained by rearranging the first order conditionassociated with the cost function. The derivation of the error correction equations is basedon the outline of the PAC framework presented in Tinsley (2002).

This section is divided into two subsections. The first presents an outline of the derivation

23

discussed in the previous paragraph, starting from a genericm-th order cost polynomial. Thesecond describes the constraint that we use for ensuring growth neutrality.

3.2.1 Cost functions and rational error-correction equations

The starting point for deriving the error correction equations is a cost function that agentsseek to minimize by choosing a sequence of values for their choice variable yt

Ct =∞∑i=0

βi

[(yt+i − y∗t+i

)2 −m∑k=1

bk

((1− L)k yt+i

)2]

(2)

The cost function penalizes deviations of the decision variable yt from its equilibrium valuey∗t (also interchangeably referred to as desired or target value) and from m differences. Theorder of the lag/lead polynomial is determined by the order of adjustment costs that theagents are assumed to be subject to, i.e. by m. In addition, β is a discount factor and bkare cost parameters (also referred to as adjustment coefficients). According to Brayton et al.(2000), differentiation with respect to yt yields

2 (yt − y∗t ) +∞∑i=0

m∑k=1

βibk∂(

(1− L)k yt+i

)2

∂yt= 0

in which∞∑i=0

βibk∂(

(1− L)k yt+i

)2

∂yt= 2bk

[(1− L)k (1− βF )k yt

]so that

(yt − y∗t ) +m∑k=1

bk [(1− L) (1− βF )]k yt = 0

which can also be expressed after some algebra as

A(L)A(βF )yt − A(1)A(β)y∗t = 0 (3)

where A(L) is a polynomial in the lag operator L and A(βF ) is a polynomial in the leadoperator F , both of degree m. With m = 1, only the change in yt is subject to adjustmentcosts with A(L) = 1 − α1L and A(βF ) = 1 − α1βF . The associated decision rule that isobtained by multiplying (3) by A(βF )−1 and significant rearranging is

∆yt = A(1)(y∗t−1 − yt−1

)+ A(1)

∞∑i=0

(α1β)i∆y∗t+i

within which terms can be rearranged and relabeled to obtain an expression that uses a morecompact notation:

∆yt = a0

(y∗t−1 − yt−1

)+∞∑i=0

di∆y∗t+i (4)

24

where a0 = d0 = A(1) and di for i > 0 are transformations of the lag/lead polynomial A andthe discount factor β.

The most ubiquitous case when this framework is applied in the FR-BDF model is wherem > 1, with m = 2 being particularly common. Then A(L) = 1 − α1L − α2L

2... − αmLm,A(βF ) = 1 − α1βF − α2 (βF )2 ... − αm (βF )m complicating the steps needed to obtain anexpression similar to (4).17 It can then be shown that when m > 1, the analogue of (4) is

∆yt = a0

(y∗t−1 − yt−1

)+

m−1∑k=1

ak∆yt−k +∞∑i=0

di∆y∗t+i (5)

with ak = −∑m

j=k+1 αj, k > 0.To a casual observer (5) might look like an error-correction equation with the additional

term∑∞

i=0 di∆y∗t+i - the discounted sum of changes in the target – with an unclear role. One

intuitive interpretation is to think of the term as a correction factor that tries to accommodatealready today for the fact that the target itself is moving and will not be the same tomorrow.If the target was expected to remain constant at its last value, the agent would not care aboutadjustment costs in the future and the sum would be zero. In the standard case this willnot be true as the target will move. Thus, the agent knows that further adjustments will benecessary in the future – even if they are exactly on target today – and that these changeswill be costly. Hence, in order to minimize total costs some additional adjustments will bereasonable even today.

Note thatdi = A(1)A(β)ι′m [1−G]−1Giιm (6)

where ιm is a m× 1 vector with the mth element equal to one, zeroes elsewhere and G is them×m matrix

G =

0 1 . . . 0 0...

... . . . ......

0 0 . . . 0 1−amβm −am−1β

m−1 . . . −am−jβm−j −a1β

(7)

3.2.2 Constraint used for ensuring growth neutrality

In order to obtain a balanced growth path in the long run, where actual variables wouldbe equal to their targets (yt = y∗t ), an important condition that needs to be met is that allequations are growth neutral. In practice this means that the sum of the ak and di coefficientsis restricted to equal unity when the model is estimated. To see why this is appropriate,assume that the system is in a balanced growth equilibrium, so that ∆yt = ∆y∗t = g.Substituting this into (5) yields

a0

(y∗t−1 − yt−1

)=

[1−

m−1∑k=1

ak −∞∑i=0

di

]g (8)

17See Brayton et al. (2000) and Tinsley (2002) for details.

25

from which it is clear that y∗t−1 − yt−1 = 0 if and only if[1−

∑m−1k=1 ak −

∑∞i=0 di

]= 0. Note

that the term∑∞

i=0 di = ω is the share of the nonstationary component of the expectedchanges in the target.

In practice, when error correction equations of the form (5) are estimated, the constraintensuring stationarity is imposed with the addition of a term resembling the right hand sideof (8) into any such equation18

∆yt = a0

(y∗t−1 − yt−1

)+

m−1∑k=1

ak∆yt−k +∞∑i=0

di∆y∗t+i +

[1−

m−1∑k=1

ak −∞∑i=0

di

]g (9)

An alternative strategy for ensuring stationarity is to re-express the expectations themselvesin terms of variables that are stationary by construction. As an example, the expectationscan be formulated in terms of deviations from long run trends. This strategy is employedoccasionally in FR-BDF, for example in the equation block used to determine corporateinvestment. See subsection 4.6.2 for details.

3.3 Computation of present values

This section describes the computation of present values under VAR-based and model-consistent expectations. We can divide expectations into two groups. Subsection 3.3.1considers the first group. These expectations appear within a PAC equation such as (5).This computation is complicated by the fact that the discounting involved depends on thecoefficients of the PAC polynomial.

The second group is considered in subsection 3.3.2. They contain present values thatare used to directly compute the value of some model variables, e.g. the long run interestrate (see subsection 4.8.2 for details). They are not affected by any frictions, and can beconstructed by a simple geometric discounting in the VAR-based case. In the MCE case,they resemble a recursive equation of a value function with a finite number of leads.

3.3.1 Present value of expected changes in the target in PAC equations

It is convenient for applications to decompose (5) into

∆yt = a0

(y∗t−1 − yt−1

)+

m−1∑k=1

ak∆yt−k +∞∑i=0

di∆y∗t+i +

∞∑i=0

di∆y∗t+i (10)

where y∗t and y∗t are the stationary and nonstationary components of the target, i.e. y∗t =y∗t + y∗t .19 This is equivalent to

∆yt = a0

(y∗t−1 − yt−1

)+

m−1∑k=1

ak∆yt−k + PV (∆y∗)t + PV (∆y∗)t (11)

18This extended version of the standard PAC equation can also be derived from a modified version of thecost minimization problem given by (2) where changes in the decision variable would be centered by g, theexogenous growth rate of the target. An additional term −b0 (∆yt+i − g)

2 would appear within the sum,implying an additional term also within the PAC polynomial.

19These components are computed with a trend-cycle decomposition, e.g. an HP filter.

26

under the definitions

PV (∆y∗)t ≡∞∑i=0

di∆y∗t+i (12)

and

PV (∆y∗)t ≡∞∑i=0

di∆y∗t+i (13)

It is clear that these infinite sums cannot be used in applied work, and that equations inwhich these sums are replaced by e.g. recursive representations are needed. There are twodifferent formulations – based on backward- and forward-looking expectations models – forhow these finite expressions are constructed.

VAR-based expectations As in Section 3.1.1, the VAR on which the backward-lookingversion of the expectations model is based can be expressed in the form of (1):

Zt = HZt−1

This equation can be iterated forward to find that for any i

Zt+i = H i+1Zt−1

i.e. we can compute projections of the components of the VAR arbitrarily far into the future,including y∗t and y∗t , if necessary. In practice, of course, the infinite sums of (12) and (13)remain impossible to operate with.

With some matrix algebra, it can be shown, however, that these infinite sums can berearranged into finite expressions. Specifically, it can be shown that there are two 1 × nvectors k0 and k1:

k0 = A(1)A(β) [(ι′mIm)⊗H ′] [Inm − (G⊗H ′)]−1[ιm ⊗ ιk0] (14)

k1 = A(1)A(β)[(ι′m (Im −G)−1)⊗H ′] [Inm − (G⊗H ′)]−1

[ιm ⊗ ιk1] (15)

where ιk0 and ιk1 are selection vectors for selecting the k0th and k1th elements of Zt whichcorrespond to y∗t and y∗t . These vectors k0 and k1 can then be applied to compute finiterepresentations of the expectations with the formulas

PV (∆y∗)t|t−1 = k0Zt−1 (16)

andPV (∆y∗)t|t−1 = k1Zt−1 (17)

Note the subscript notation on the right hand side of equations (16) and (17). With theindices t and t − 1 we wish to denote the fact that the computation of the discounting inthe sum is done with respect to period t and that the future terms y∗t+i and y∗t+i, i > 0 areconstructed using the information set of period t− 1.20

20This extra notation is unnecessary under MCE since, for agents with perfect foresight, the informationset contains everything.

27

Model-consistent expectations The derivation of the finite expressions in the model-consistent case, with an information set ending in t, starts by multiplying (3) by A(βF )−1

to obtainA(L)yt = A(βF )−1A(β)A(1)y∗t + A(βF )−1A(β)A(1)y∗t (18)

which has exactly the same form as (5), with the exception that all terms involving yt arenow on the left hand side. Thus the first term on the right hand side must be equal toPV (∆y∗)t + A(1)y∗t−1, and the second term to PV (∆y∗)t + A(1)y∗t−1.

Thus for the first term we have (noting that A(1) = a0)

PV (∆y∗)t = −a0y∗t−1 + A(βF )−1A(β)a0y

∗t

which is equivalent to

A(βF )PV (∆y∗)t = a0

[A(β)y∗t − A(βF )y∗t−1

]resulting in

PV (∆y∗)t = −m∑i=1

αiβiPV (∆y∗)t+i + a0

[∆y∗t +

m−1∑k=1

(m−1∑j=k

αj+1βj+1

)∆y∗t+k

](19)

Due to symmetry, an equivalent equation

PV (∆y∗)t = −m∑i=1

αiβiPV (∆y∗)t+i + a0

[∆y∗t +

m−1∑k=1

(m−1∑j=k

αj+1βj+1

)∆y∗t+k

](20)

holds also for PV (∆y∗)t, i.e. the trend component of expectations. These two equations –(19) and (20) – can be conveniently used in model-consistent applications instead of (16)and (17) that are used in VAR-based applications.

3.3.2 Present values with a constant discount factor

In some cases such as asset pricing equations, present values are discounted with a constantdiscount factor based on the expectation of the following sum:

(1− β)∞∑i=0

βiyt+i (21)

Under VAR-based expectations, with an information set ending in t− 1, we compute thepresent value with the following formula:

PV (y)t|t−1 = (1− β)ι′

h(I − βH)−1HZt−1. (22)

Under model-consistent expectations, with an information set ending in t, we computethe present value with the following recursive formula:

PV (y)t = βEtPV (y)t+1 + (1− β)yt. (23)

28

4 Model specification and estimationThis section describes the full model in detail. After describing the notation used, we proceedblock by block, starting with supply and ending with public finances and the accountingframework. In what follows the equations that we present are written for the VAR-basedcase.

4.1 Estimation approach

As the model is estimated block by block, there is no need for a universal estimation method-ology –the estimation methods of FR-BDF are highly dependent on the particular featuresof individual model components. However, given some interdependence between the variousmodel components, there is some structure to the overall estimation procedure. The modelwas estimated using French Quarterly National Accounts in base 2014, from 1995-Q1 to2017-Q4.21 For the calculation of capital services (see section 4.3), we used wealth accountsin base 2010 and re-based capital and investment sub-components in 2014.

Many of the equations describing short run dynamics include expectation terms whichrequire estimating a model to compute these terms. The model we apply is E-SAT. Giventhat E-SAT depends on the output gap, the estimation process is as follows: (i) we calibratethe production function consistently with estimated factor demand conditions and the factor-price frontier (see section 4.3) and obtain an estimate of the output gap, (ii) we estimateE-SAT using Bayesian methods and (iii) we estimate the rest of the model equations, usingE-SAT to compute expectations in PAC equations.

The model equations are mostly estimated using a methodology that closely follows thatof FRB/US, as the so called short run equations are all estimated with iterative OLS. Thecore equations of the backward-looking expectations model are estimated with Bayesianmethods, while all auxiliary equations needed for expectations formation are estimated withOLS. All long run equations describing targets are estimated with simple OLS. Finally,some coefficients are calibrated, such as the production function parameters (except thelabor/capital elasticity of substitution estimated as a parameter of the long run equation ofbusiness investment) and the markup of monopolistic firms.

The iterative estimation of the short run equations of the model is based on OLS: aninitial guess is made on the PAC coefficients of the equation, which can be used to computea discounted sequence of expectations using E-SAT, which is in turn used as an observable inthe estimation of the PAC coefficients. Given these new estimates, the expectations sequencecan be recomputed, the PAC coefficients re-estimated and so forth until convergence.

The estimation of equations following the PAC framework requires an assumption to bemade regarding the discount factor β appearing in the cost minimization problem expressedin (2). We follow the approach of FRB/US and calibrate this number to be 0.98 in allmain blocks of the model except household consumption (see section 4.6.1 for details). Thisdiscount factor is consistent with the real rate of return for financial assets of roughly 8%

21We used the 2018-Q1 QNA detailed figures published on 22 June 2018. Estimation samples can varydepending on: (i) equation specifications, (ii) data availability for variables that are external to QNA and(iii) modellers’ judgment.

29

observed in the US during the postwar period (Brayton et al., 1996) and in France duringthe period 1970-2014, as shown by Garbinti et al. (2017).

It is important to note that the model is estimated with the VAR-based expectations forthe reasons explained in the bird’s-eye-view, see section 2.8. When FR-BDF is simulatedwith model-consistent expectations, it means that only the expectations are changed, butnot the coefficients of the equations in which they enter. Take for instance the wage Phillipscurve. The Phillips slope is estimated as an elasticity of wage inflation with respect to theexpected discounted sum of future unemployment gaps. The latter is the policy function ofthe E-SAT variables in t− 1. In the MCE case, expectations take on a forward-looking formbut the slope remains unchanged, because we assume that on a "historical" sample theseexpected sums are the same in both cases.

4.2 Notation

While the model notation in code follows very closely the standard notation laid out in theSystem of National Accounts (United Nations, 2009), we prefer here to use shorter notationsin order to present more compact formulas. In addition, given that FR-BDF is a very largemodel that e.g. makes strong distinctions between agents and sectors and contains explicitlyprices, volumes and values for the same economic concepts while using certain complexmathematical transformations, it is necessarily complexity and somewhat opaque. In thissubsection we attempt to clarify these issues.

Our notation introduces a number of operators that are particularly common in FR-BDF.The first is the expectations operator, which we denote by PV (x)t|t−k where xt is a modelvariable. The subscript describes the timing of information: the first component t refers tothe date when the expectation is constructed, while the second component t − k refers tothe information set available for the construction. The second is the gap, by which we meanthe deviation of xt from its long run trend xt. This gap is denoted by xt, i.e. xt = xt − xt.

Furthermore, we follow some typographical conventions in order to simplify our notation.Lower-case letters will be used to denote logarithms – e.g. the logarithm of householdconsumption Ct will be ct – or interest rates, e.g. the short rate will be denoted it and theweighted average cost of capital wacct. Other rates or ratios will be denoted with the letterτ .

Finally, Table 4.2.1 presents, for the sake of convenience, the variables and notation forthe core variables appearing in the expectations satellite model E-SAT, which is transverseacross FR-BDF.

4.3 Supply block

FR-BDF has a fully specified supply block which enables the model to endogenously convergetoward the long run or natural level of GDP (see section 5.1), as well as to define the outputgap used in E-SAT. The model is based on a standard neoclassical growth model to whichwe add monopolistic competition.

We model separately the market branches and the non-market branches.22 Market22Market branches group agriculture (AZ in Insee codification), industry (DE and C codes), construction

30

Table 4.2.1: E-SAT core variables

Notation Description

yt Output gap, in deviation from the long run outputit Short run interest rate, 3-month Euriborit Long run trend of the short run interest rateπQ,t Value added price inflation of market branchesπQ,t Long run trend of the value added price inflation of

market branchesyEA,t Euro area output gap, in deviation from potential

outputπEA,t Growth rate of the euro area GDP deflatorπEA,t Long run trend of the GDP deflator inflation

branches’ supply is based on a micro-founded theoretical framework with a production func-tion technology and equilibrium conditions derived in absence of adjustment costs. In con-trast, for non-market branches output, we do not assume any production function technology.This is furthermore justified by the fact that these variables are basically exogenous in con-ditional forecasting; in unconditional simulation, they are assumed to grow at the same rateas long-run real GDP.

This section presents the theoretical framework from which we derive the specificationof desired targets for business investment, employment and value added price.23 We thenpresent the "model-consistent" calibration of FR-BDF’s production function of firms. Fi-nally, we define the level of long run output toward which FR-BDF converges endogenously.

4.3.1 Theoretical framework of market branches’ supply

At the level of market branches, the production function technology is assumed to be aConstant Elasticity of Substitution (CES) production function, rather than a Cobb-Douglasproduction function. This makes it possible to have a more plausible non-unitary elasticity oflabor and investment to labor cost and capital user cost. In addition, we are able to preservethe Balanced-Growth Path hypothesis by assuming a labor-augmenting technical progress.In this subsection, we derive equilibrium conditions for capital services, employment andvalue added price in the absence of adjustment costs.

Production function technology Market branches’ aggregate value added Qt is deter-mined by the following CES production function

Qt = F (Kt, HtNt, Et) ≡ γ[αK

σ−1σ

t + (1− α) (EtHtNt)σ−1σ

] σσ−1

(24)

(FZ code) and services (codes going from GZ to MN). Non-market branches correspond to the branch ofnon-market services (OQ code).

23The estimation of corresponding short run equations and the properties of these blocks will be detailedin sections 4.6.2, 4.5.2 and 4.4.

31

Table 4.3.1: Variables used in Section 4.3

Notation Description

Qt Value added of market branches, volumePQ,t Value added deflator of market branchesKt Capital services, market branches excluding agricultural and

real estate branches, volumeIt Investment, market branches excluding agricultural and real

estate branches, volumeδt Depreciation rate of capitalNt Total employment of market branches, thousands of personsNS,t Salaried employment of market branches, thousands of personsHt Working time per capita in market branches, hoursEt Solow Residual of market branchesEt Trend labor efficiency (labor-augmenting technological

progress)Wt Total labor cost per worker, value, market branchesrK,t Real user cost of capital, excluding agricultural, real estate and

public sectorwacct Weighted average cost of capitalπQ Value added price inflation at time t conditional on information

at time t− 1, using E-SATQ′K,t Marginal product or return on capital, volumeuN,t Long run equilibrium level of unemployment, in percentageψt HP-trend share of market branches employment in total em-

ploymentPopt HP-trend of labor force, in thousands of personsQnmt Non-market branches GDP, volumeYt Total economy GDP, volumeuN,t Long run equilibrium level of unemployment, in percentageQN,t Market branches’ long run value added, volumeYN,t Total economy long run GDP, volume

Note: tilde-marked variables are specific to this section. We use it to make a distinctionbetween total business investment of market branches It and total business investmentexcluding agricultural, real estate and public branches investment It; the same appliesto real user cost of capital services rK,t. Wt denotes total labor cost, i.e. gross wagesplus employers’ social contributions.

32

where Kt is a measure of capital services, NtHt denotes labor supply measured in total hoursand Et a labor-augmenting technical progress;σ is the elasticity of substitution between laborand capital.24 Et is obtained as a Solow residual by inverting the production function (24)such that:

Et =

[(Qtγ

)σ−1σ − αK

σ−1σ

t

(1− α)(HtNt)σ−1σ

] σσ−1

(25)

Input demand and factor price frontier Under monopolistic competition, firms maxi-mize profit by setting a constant optimal markup µ. Profit maximization yields the followingstandard first order conditions, determining the long run equilibrium levels of capital andlabor demand, which equate marginal productivity and markup-augmented marginal costfor each factor:

rK,tPQ,t

=α

µγσ−1σ

(Qt

Kt

) 1σ

(26)

Wt

PQ,t=

1− αµ

γσ−1σ EtHt

(Qt

EtHtNt

) 1σ

(27)

The real user cost of capital services is defined as:

rK,tPQ,t

=(wacct + δt − PV (πQ)t|t−1

) PI,tPQ,t

(28)

where PV (πQ)t|t−1 is the expected present-value of VA price inflation at time t conditionalon information at time t − 1 obtained from E-SAT, wacct is the weighted average cost ofcapital and PI,t/PQ,t is the relative price of business investment to VA price. Our measureof the real user cost of capital implicitly assumes no adjustment costs; see section 4.6.2 formore details.

Hence, using production function (24) and labor demand (27), we derive the followingFactor Price Frontier (FPF, henceforth):

PQ,t =µ

γ(1− α)

σ1−σ

[1− ασ

(Q′K,tγ

)1−σ]− 11−σ Wt

EtHt

(29)

where marginal productivity or return on capital is defined by the partial derivative ofproduction function (24) with respect to Kt

Q′K,t ≡ α (γ)σ−1σ

(Qt

Kt

) 1σ

(30)

This non-standard version of the FPF calls for some comments. In structural macroe-conomic models, the standard FPF generally relates the output price to the labor cost and

24In the measurement of Kt, we exclude agricultural (AZ code), housing services (LZ code) and non-market services (OQ code) capital to focus on productive capital. Capital services are computed usingnational wealth accounts with a methodology close to Cabannes et al. (2013).

33

the user cost of capital. Such a relation is usually derived using both capital and labordemand conditions, i.e. assuming both capital and labor markets are at equilibrium. Af-terwards, because of adjustment costs on investment in FR-BDF (see section 3.2), capitalwill be inelastic, while the real cost of capital RK,t is computed under the assumption of theabsence of adjustment costs (i.e. expected capital gains are assumed to be equal to expectedinflation, see section 4.6.2). Hence, we do not rely on condition (26) that the capital marketis at equilibrium when deriving the FPF. As a result, firms’ pricing behavior depends onthe marginal return on capital rather than on the real user cost of capital. We interpretthis specification of the FPF as a resource constraint with the labor market at its long runequilibrium (27).

4.3.2 Equilibrium of the supply of market branches

Specification FR-BDF’s supply block is derived from equations (27), (26) and (29) whichact as a resource constraint. We now distinguish equilibrium variables (K∗t , N∗t , P ∗t ) fromactual variables (Kt, Nt, Pt) denoting the first ones with stars. These equilibrium variablesdefine the targets of behavioral short run equations, which take into account the presenceof adjustment costs (presented in the following sections). In practice, we slightly deviate insome cases from the theoretical equations derived earlier, in order to provide a better fit tothe data.

First, following FRB/US, we transform the capital demand condition (26) into an invest-ment demand equation. In order to do so, we define the equilibrium capital K∗t by

K∗t ≡(α

µ

)σγσ−1

(rK,tPQ,t

)−σQt (31)

Then, we evaluate the equilibrium investment from the steady-state of the capital accumu-lation equation:

Kt = (1− δt)Kt−1 + It (32)

where δ is the depreciation rate of capital and It is the investment of market branches,excluding agricultural and housing services. We model capital services with such an aggregatecapital accumulation formula for the sake of simplicity, although capital accumulation occursin national accounts at the level of each asset.25 Let denote the trend growth rate of capital bygKt measured by the HP filter and rewrite (32) to get the definition of equilibrium investment:

I∗t ≡δt + gKt1 + gKt

K∗t (33)

Then, replacing K∗t by (33) in (31) yields an optimal demand for aggregate market branchesinvestment I∗t :

I∗t ≡δt + gKt1 + gKt

(α

µ

)σγσ−1

(rK,tPQ,t

)−σQt (34)

25In our computation of capital services, for each asset, capital services are assumed to be proportional tothe net capital stock of this asset, which has in national accounts its specific depreciation rate.

34

Finally, the equilibrium investment target is directly derived from (34) by taking logs:

log I∗t = a0 + log(Qt)− σ log

(rK,tPQ,t

)+ log

(δt + gKt1 + gKt

)(35)

Second, the Solow residual Et is replaced by trend labor efficiency Et in each equation.This choice is motivated by the fact that the Solow residual is volatile and combines bothcyclical and structural factors, which are unrelated to the labor-specific technical progress.In particular, the Solow residual does not take account of the utilization rate of productioncapacity. As a result, we assume equilibrium demand conditions for labor and investmentas well as the evaluation of FPF at the trend level of labor efficiency Et. Trend efficiency isdefined as the level of efficiency with a utilization rate of production capacity equal to its longrun mean and estimated with the following three assumptions: (i) Et follows a deterministictrend with multiple breaks in its slope, (ii) there is a break in level (or "step") in 2008-Q3to account for a permanent effect of the 2008-09 recession and (iii) there is an autoregressivestructure to account for a smooth transmission of shocks (e.g. the 2008-Q3 step). Figure 4.3.1represents both the Solow residual and the estimated trend labor efficiency whose annualgrowth rate is estimated at 2.4% before 2002-Q2 and at 0.85% afterwards. The 2008-Q3 stepis estimated at 4.2%.

Third, we choose to estimate a labor-demand equation for salaried employment insteadof total employment of market branches26. This choice is mostly due to a better econometricperformance of the salaried employment equation (see section 4.5.2 for details). However,in order to bridge our theoretical model with the estimated equation for labor demand, wewould need to recover the estimates of the total employment equation. Taking logs of (27),we obtain:

log(N∗t ) = b0 + log(Qt)− log(Et)− σ logW

PQ,tEt+ (σ − 1) log(Ht) (36)

Then define νt = NS,t/Nt. Note that the OLS estimate of b0 is equal to b0 − log(ν) wherelog(ν) is the empirical mean of log νt and b0 is the estimated intercept of the following salariedemployment equation:

logN∗S,t = b0 + log(Qt)− log(Et)− σ logW

PQ,tEt+ (σ − 1) log(Ht) (37)

As a result, we can recover the estimate for b0 from the intercept b0 and from log(ν) stillwithout directly estimating equation (36).27

Fourth, we use a measure of trend return on capital Q′K,t rather than observed returnQ′K,t. The trend is obtained by using an HP filter on observed Q′K,t in the historical samplebut is projected according to an exogenous AR(1) process, anchored to the steady-state of

26Indeed, equation (27) defines the demand for total employment. Within our sample (1995Q1-2017Q4),the evolution of the share of salaried employment in market branches has been small: it rose from around86% in 1995Q1 to a peak around 89% in the mid-2000s and returned to 88% in 2017Q4.

27This is notably due to the fact that the only estimated parameter is the intercept, while σ is calibrated.

35

Figure 4.3.1: Solow residual and trend labor efficiency

.09

.10

.11

.12

.13

.14

.15

.09

.10

.11

.12

.13

.14

.15

1990 1995 2000 2005 2010 2015

Solow residual Trend labor efficiency

equation (26) (see section 4.9). Hence, for given values of σ and α the estimated equationfor the target of VA price equilibrium is:

log(P ∗Q,t) = c0 +σ

1− σlog(1− α)− 1

1− σlog

[1− ασ

(Q′K,tγ

)1−σ]+ log

Wt

EtHt

(38)

Figure 4.3.2 shows the marginal return of capital Q′K,t and the trend marginal return ofcapital Q′K,t, both annualized. First, the annual marginal return of capital fluctuates between30% and 22%. This value for Q′K,t is broadly in line with the level of the real user cost ofcapital augmented by the markup.28 Second, we observe a downward trend in the marginalreturn of capital because capital services have grown faster than real output since the early2000s.29

Calibration We now turn to the calibration procedure of the production function. Welabel it "model-consistent calibration" since the production function’s parameters (α, σ andγ), markup (µ) and trend labor efficiency are calibrated consistently with estimated equilib-rium conditions (35), (37) and (38).30 Indeed, the calibration of the production function, the

28The markup µ is calibrated at 1.31. The annualized real user cost of capital fluctuates between 21%and 18%. Regarding its subcomponents, the WACC fluctuates between 8% and 5% (see Figure 4.8.2), thedepreciation rate is equal to 15% per year on average between 1995 and 2017 and expected inflation is around2% (annual rate).

29Marginal return of capital Q′K,t is homogenous to the output-to-capital ratio Qt/Kt, see eq. (30) andtherefore its growth depends on the relative growth of real output and capital services.

30Contrary to other authors such as Klump et al. (2012), we neither normalize the CES production functionnor calibrate it with big ratios (i.e. capital share, labor share, capital/output ratio), because the resulting

36

Figure 4.3.2: ObservedQ′K,t and trendQ′K,t annual marginal return of capital, in percentage

20

22

24

26

28

30

32

20

22

24

26

28

30

32

2000 2005 2010 2015

Marginal return of capital Trend marginal return of capital

estimation of labor efficiency and estimates of equilibrium equations for business investment,employment and VA price are mutually dependent.

From equation (25), the calibration of the production function’s parameters (α, σ and γ)determines the level of both the Solow residual and trend labor efficiency, which determinestarget labor demand and VA price equations. In turn, intercepts of the three equilibriumequations – investment and labor demand and VA price – are non-linear functions of theseparameters and of the markup µ. Let (a0, b0, c0) be respectively the three estimated interceptsof equations (35), (37) and (38). If the production function and markup were estimatedconsistently with these equations, we should meet the following cross-restrictions betweenestimated and theoretical intercepts:

a0 = log

[(α

µ

)σ(γ)σ−1

](39)

b0 = log

[(1− αµ

)σ(γ)σ−1

]+ log ν (40)

c0 = log

[µ

γ

](41)

where the theoretical intercepts are the right-hand side terms. Since FR-BDF is estimatedequation-by-equation, applying cross-restrictions without a joint-estimation of these equa-tions is challenging. Hence, we developed a strategy to estimate FR-BDF sequentially,without departing from the equation-by-equation estimation strategy.

calibration would not necessarily be consistent with estimated behavioral equations, where cross-restrictionswould not have been imposed.

37

As a first step,we obtain an estimate for the capital-labor elasticity of substitutionσ = 0.53 from the firms’ target investment I∗t equation(63), which is independent of thetrend efficiency estimate and has quite a similar structure to equation (35), albeit withsome differences (see section 4.6.2 for more details). In this step, we specifically assumePV (πQ)t|t−1 = πt and evaluate rK,t in equation (63) at the long run anchor of inflationexpectations, because E-SAT has not been estimated at this stage. We do not estimate σusing It since it is not available from quarterly national accounts and for which we constructa quarterly measure from annual data by interpolation.

As a second step, we obtain an estimate of the investment equation’s intercept a0. At firstglance, we have a system of three equations (39), (40), (41) and three unknowns (α, γ, µ),which we would solve by grid search. However, it can be reduced to a two equation–twounknown system since business investment equation (35) does not depend on trend efficiency.Consequently, the investment equation’s intercept a0 can be estimated outside the grid searchand we can deduce markup µi as a function of estimated a0, calibrated σ and grid parameters(αi, γi) from the following equation:

µi = µ(αi, γi, σ, a0)

= exp

(logαi +

σ − 1

σlog γi −

a0

σ

)(42)

Similarly to rK,t in the first step, we evaluate rK,t at the long run anchor of inflation expec-tations for the reasons exposed above.

As a third step, we use a grid search to obtain the calibration of the FR-BDF productionfunction that is consistent with estimated intercepts, with σ being calibrated. We run thegrid search for plausible but sufficiently large ranges of α and γ which we expect to be in[0.2, 0.4], with a step parameter equal to 0.001. Consequently, we have exactly 201 pointsfor each parameter, which results in 40401 grid points. Hence, for each point (αi, γi) of thegrid (α, γ)2, the grid search proceeds as follows:

1. Calculate µi and Q′K,t using equation (30).31

2. Compute the Solow Residual Et from equation (25) and estimate trend labor efficiencyEt with the method described above.

3. Estimate b0 and c0 by OLS from equations (37) and (38), for a given Et and forparameters (σ, αi, γi, µi).

4. Define the following `1-norm:

||x||1 :=

∣∣∣∣b0 − log(1− αi

µi

)σ(γi)σ−1 − log ν

∣∣∣∣+

∣∣∣∣c0 − logµiγi

∣∣∣∣31In practice, we do not estimate the VA price equation using the HP-trend return on capital Q′K,t but

with Q′K,t. Since the only estimated parameter is the intercept, the OLS estimator of c0 only depends onregressors’ empirical means. Thus, using the observed or HP-trend return on capital is perfectly equivalentbecause they have the same historical mean.

38

5. Finally, pick the grid point which minimizes ||x||1:

(α, γ) = arg minαi,γi

||x||1

provided min||x||1< 10−3 to obtain µ according to equation (42).

Table 4.3.2 summarizes the model-consistent calibration we obtain and compares it to whatwe would obtain with the normalization of the CES and big ratios. The calibration that weobtain is consistent with estimated equilibrium conditions (business investment, employmentand VA price) and differs slightly from the big ratios method, especially for markup µ andscale parameter γ.

Table 4.3.2: Calibration of the FR-BDF production function

σ α γ µ min||x||1

Model-consistent calibration 0.53 0.26 0.34 1.31 0.0006Calibration using big ratios 0.53 0.26 0.29 1.42 -

The model-consistent calibration is critical for the long run convergence toward the longrun levels of GDP YN and sector value added QN (see section 4.3.3). In contrast, if wewere calibrating the production function using the method of big ratios, the only way forthe VA to converge toward its long run level would be to calibrate the long run equilibriumequations’ intercepts (a0, b0, c0) to their theoretical values. Such a strategy would inevitablyworsen the empirical fit of estimated equations. In addition, normalization and big ratioswould require that the economy be on a balanced growth path in the historical sample –which is not verified, in particular regarding the capital/output ratio.

4.3.3 Long run output in FR-BDF

We now present the long run output of FR-BDF. As in the traditional approach of potentialoutput, we define here long run output as the level of output that can be achieved for a givenlevel of capital services. We use a specific name for this object, because some assumptionshave been made specifically to be consistent with the long run of FR-BDF, while the Banquede France estimates of potential output do not necessarily adopt these constraints.

In contrast, the concept of long run output in FR-BDF differs from the one usually foundin DSGE models. In FR-BDF, rigidities, whether real or nominal, are captured by the costsof adjusting variables toward their targets. The long run output is defined by the state of theeconomy in which both nominal and real rigidities have waned, except for capital servicesconsidered as given. On the contrary, in DSGE models, long run (or natural) output isgenerally defined as the flexible-price level of output, i.e. the level of output in the absenceof nominal rigidities but given real rigidities.

As mentioned earlier, market and non-market branches are treated differently and sepa-rately. The long run output of market branches is derived from a production function, whilenon-market branches’ long run output is defined by a statistical trend. Finally, the longrun output of the total economy is calculated using a chained price aggregation of these twosub-components.

39

Long run output of market branches The long run output of market branches isderived from the production function given exogenous trend of the labor force, trend efficiencyand given actual level of capital services.32 Hence, the long run level of sector value addedis defined by:

QN,t = γ[αK

σ−1σ

t + (1− α)(EtHtψt(1− uN,t)Popt

)σ−1σ

] σσ−1

(43)

where Ht denotes the HP-trend hours per worker. Since long run equilibrium unemploymentuN,t is estimated on the total economy employment instead of total employment, we need todefine ψt as the trend share of market employment in total employment, in order to recoverthe trend of total employment. As a result, Htψt(1 − uN,t)Popt represents trend of laborsupply (in hours).

Besides trend efficiency Et (see above) and long run unemployment uN,t which havespecific estimates, both trend hours and trend share of employment in total employmentare obtained from a standard HP filter with λ = 1600 on quarterly data.33 The long rununemployment rate is estimated outside FR-BDF by Kalman filtering, as a time-varyingintercept of a price-inflation Phillips curve.34 Finally, we define the market branches’ outputgap by Qt ≡ Qt/QN,t − 1.

Long run output of non-market branches First, we define the output non-marketbranches Y nm

t as the chained price difference between GDP (Yt) and market branches’ valueadded (Qt), which includes the value-added of public and non-profit institutions servinghouseholds (NPISH) sectors but also taxes and subventions on products. Hence, the longrun output of non-market branches Y nm

t is defined by a statistical trend and is projected suchthat the non-market output gap will close mechanically, like other trend variables in FR-BDF (see sections 4.9). In practice, we measure the non-market branches’ trend GDP usingthe HP filter with a smoothing parameter λ = 1600. In future developments of FR-BDF,we could introduce an explicit production function technology for the non-market branches,which is mostly the public sector.

Aggregation and total economy long run output Finally, the long run output of thetotal economy YN,t is obtained by chained-prices aggregation of QN,t and Qnm

t and the outputgap of the total economy is denoted by Yt ≡ Yt/YN,t − 1. At the balanced-growth path, thelong-run output of total economy YN,t as well as its sub-components QN,t and Qnm

t growat a constant annual rate ∆yN , which is the sum of the growth rate of labor efficiency ∆eand the growth rate of labor force ∆¯. We estimate the trend annual growth-rate of laborefficiency at ∆e = 0.85%, see section 4.3. Regarding labor force, we assume a trend annualgrowth-rate equal to ∆¯ = 0.2%, which is the average growth-rate of labor force over the

32We use the actual level of capital services because we assume that capital does not deviate much fromits long run level.

33The trend share of market employment is quite stable over the sample: it increases from around 69.8%in 1996Q1 to 71.2% in 2002Q4 and returns to 70.2% in 2017Q4.

34This estimate is provided by the external trade and structural policies studies unit of Banque de France.Another approach left for further research could consist in calculating an equilibrium unemployment ratedirectly from the wage Phillips curve of FR-BDF.

40

period 2015-2045 based on Insee labor force projections for 2070 published in 2017.35 Giventhese technical assumptions, the long-run output growth-rate ∆yN is equal to 1.05% at thebalanced-growth path.

4.4 Value added price of market branches

The value added price equation is one of the key equation in FR-BDF since this deflatorenters the equations of all the other types of prices. It enables expectations to affect pricesetting.

Table 4.4.1: Variables used in section 4.4

Notation Description

Q′K,t Hodrick-Prescott filter of marginal product orreturn on capital, volume

Ht Working time per capita (in hours)Wt Total labor cost per capita, gross wages plus em-

ployers’ social contributions, valueEt Long run efficiency

∆et Efficiency trend, ∆log(Et)pQ,t Value added price of market branch (in log)p∗Q,t Target of the value added price (in log)π∗Q,t Hodrick-Prescott filter of π∗Q,tπt Long run anchor of inflationπQ,t Value added price inflationyt Output gapπW,t Growth rate of wage per capita of market

branchesPV (π∗Q)t Present value of the expected growth rate of the

value added price targetπ Long run value of inflationL Lag operator

E-SAT See Table 4.2.1πW,t Efficient wage inflationut Unemployment gap

Target The long run of the VA price is derived from the price frontier. It is given byequation (38) described in detail in section 4.3.1, where the price target depends on themarginal productivity of capital (instead of its user cost) and on the efficient hourly labor

35Insee projections are available at: https://www.insee.fr/fr/statistiques/2844302.

41

cost. For the sake of convenience we provide it below:

p∗Q,t = c0 +σ

1− σlog(1− α)− 1

1− σlog

[1− ασ

(Q′K,tγ

)1−σ]+ log

Wt

EtHt

All parameters except the constant are calibrated or estimated within other equations,see Table 4.4.2.

Table 4.4.2: Estimates and calibrated parameters, value added price, long run

Coef. s.e.

σ 0.53 -α 0.26 -µ 0.335 -γ 1.31 0.001R2 = 0.97

Short run equation The short run is estimated using the PAC framework on the 1997Q1-2017Q4 sample. We added a direct effect of current demand in the short run which issupposed to capture in a reduced form the behavior of non-optimizing firms. The parameterestimates are presented in Table 4.4.3.

πQ,t = PV (π∗Q)t|t−1 + β0

[p∗Q,t−1 − pQ,t−1

]+ β1πQ,t−1 + β2yt

+(1− β1 − ω)π∗Q,t−1 + εt(44)

Table 4.4.3: Estimates and calibrated parameters, value added price, short run

Coef. s.e.

β0 0.06 0.02β1 0.50 0.09β2 0.09 0.03ω 0.46 -

R2 = 0.40

Expectations In order to build expectations of the growth rate of the value added pricetarget we had to include three additional equations into the core of the E-SAT model. Thefirst one is to describe the change in the target itself. Since we are constrained in the numberof auxiliary equations that we can add, the easiest way to model the growth rate of the valueadded price target is to link it to the real efficient wage, which was actually used to constructthis target36:

π∗Q,t = β0(πW,t −∆et) + (1− β0)π∗Q,t + εt (45)36Note however that here we are using net wage rather than gross wage as in the target equation.

42

It would be logical to use πt as a trend instead of π∗Q,t but then we would need a constantto center the residuals.37 The second equation is a Phillips curve, linking the real effectivewage to the unemployment gap:

(1− ρL)

[πw,t −∆et − π∗Q,t

]= β0ut + εt (46)

where L is a lag operator.The third equation mimics an Okun’s law, relating the unemployment gap to the output

gap with an AR(1) process in residuals:{ut = β0yt + ηt

ηt = ρηt−1 + εt⇒ ut = β0(yt − ρyt−1) + ρut−1 + εt (47)

The estimated parameters are available in Table 4.4.4. As was already mentioned, wedid not find a high R2 of auxiliary equations, but for meaningful coefficients. β0 from theOkun’s law in equation (47) confirms our prior, see discussion in subsection 4.5.1. A negativecoefficient in front of yt−1 in the policy function is offset by a positive coefficient with respectto ut−1. The elasticity of the value added price with respect to the output gap is, in the endpositive: πQ,t

yt−1= (−0.15 + 1.1 · 0.246) · 10−2 = 1.3 · 10−3 > 0. The slope of the Phillips curve

measured by the ratio β01−ρ from equation (46), is equal to 0.53, which is a bit higher than the

one computed from a partial equilibrium of the wage Phillips curve (see equation (52)) butstill lies in the boundaries of commonly accepted values. A high elasticity of PV (π∗Q)t|t−1

with respect to π∗Q,t in the policy function is related to a constraint of the balanced growthpath condition of equation (45). However, it has no implication for the forecasting exerciseor policy experiments since π∗Q,t is exogenous in the model:

π∗Q,t = α0π∗Q,t−1 + (1− α0)π (48)

where α0 is calibrated to 0.95 (which corresponds to a half life of 12 quarters) and π is 0.0048.

Dynamic contributions Figure 4.4.1 describes how the various terms of the long andshort run equations of the value added price have contributed to variation in its growth rate.Almost all positive dynamics of the latter are explained by the value added inflation target.Expectations are as important for the dynamics of the value added price inflation as for theoutput gap. After the second half of 2009, the expectations put a downward pressure onthe value added inflation. Indeed, these expectations are strongly influenced by the trendinflation of the price target, which is subdued in the post-crisis period.

37As explained above, πt is higher than the mean of the value added price over our estimation sample.

43

Table 4.4.4: Policy function of the growth rate of the value added price target

Regressors Policy function Aux. equation Aux. equation Aux. equationPV (π∗Q)t|t−1 π∗Q,t (eq 45) πW,t (eq 46) ut (eq 47)

E-SAT Model variablesyt−1 −1.5 · 10−3 - - 0.23it−1 − it−1 −3.4 · 10−3 - - -πQ,t−1 − πQ,t−1 8.7 · 10−4 - - -yEA,t−1 4.8 · 10−4 - - -πEA,t−1 − πEA,t−1 0.00 - - -ut−1 −1.1 · 10−2 - - 0.946 [0.04]πW,t−1 1.2 · 10−2 - 0.25 [0.1] -π∗Q,t−1 0.44 - -0.25 [0.1] -yt - - - −0.246 [0.07]ut - - −0.04 [0.05] -πW,t - 0.59 [0.09] - -π∗Q,t - 0.41 [0.09] - -

R2 = 0.48 R2 = 0.02 R2 = 0.92Note: standard errors in brackets. πW,t = πW,t −∆et.

Figure 4.4.1: Dynamic contributions, Value added price quarterly inflation, in pp

-0.8

-0.4

0.0

0.4

0.8

1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17

Value added price target Expectation of the change in the value added price target

Hodrick-Prescott filter of value added price target Output gap

Residuals Value added price inflation

44

4.5 Labor market

The labor market block includes (i) labor supply: the wage Phillips curve which defines thelink between wage inflation and the expected unemployment gap; and (ii) labor demand:the employment equation, which relates employment to labor costs and aggregate demandin the presence of adjustment costs.

4.5.1 Labor supply

Table 4.5.1: Variables used in section 4.5.1

Notation Description

wt Log of gross wage per head of market brancheswmt Log of gross minimum wageπW,t Wage inflationπC,t Growth rate of the consumption deflatorpC,t Log of the consumption priceet Log of the long-run efficiency∆e The long run anchor of the efficiency growth rate

PV (u)t|t−1 Present value of the expected average of future un-employment gaps

ut Unemployment rateuN,t Long-run trend of the unemployment rateπt Long-run anchor of inflation

E-SAT See Table 4.2.1ut Unemployment gap

In the long run the labor supply curve is vertical: the wage elasticity of supply is zero.The unemployment rate is anchored to an exogenously defined long run level, uN,t. Asexplained in subsection 4.3.3, we measure this variable using an estimate provided by theTrade and Sstructural Policies Analysis Division of the Banque de France.

In the short run, the labor supply in FR-BDF is defined by a wage Phillips curve.38 Theequation is augmented with hybrid indexation, which is an important element to recover asignificant role for the expected unemployment gap. The wage equation is microfounded.It is derived from the first order condition of agents’ optimization problem with respect toleisure. It is important to bear in mind that as FR-BDF is a semi-structural model, thewage equation is not derived jointly with the consumption/saving decision, i.e. we do notimpose cross-restrictions between coefficients. We follow Gali et al. (2011), and consider thefollowing New Keynesian wage Phillips curve with indexation:

πW,t − xt−1 = α + βEt−1 [πW,t+1 − xt]− λ(ut − uN,t) (49)38We also considered a wage setting approach but obtaining a significant coefficient of adjustment toward

the long run proved difficult.

45

where the variable xt−1 captures the indexation of households unable to optimize their wagein the current period. This indexation variable is determined by the following equation:

xt−1 = c1πC,t−1 + c2[πW,t−1 − c1πC,t−2] (50)

For notations see Table 4.5.1.Along with indexation, we also add a real efficient minimum wage, computed as the

minimum wage per capita adjusted for labor efficiency and long run anchor of inflation. Itenters the equation as a year-on-year growth rate (∆4) in order to solve the seasonality issue,see Figure 4.5.1. The minimum wage centered by its trend (πC,t + ∆et) is endogenized asfollows:

wmt = δq1

(π4,C,t + ∆4et + 0.5 [π4,W,t − π4,C,t −∆4et]

)+ εt (51)

where δq1 is a dummy that takes on a value of 1 in the first quarter of each year; π4,C,t isyear on year consumer price inflation and π4,W,t is year on year wage inflation. Similarly tothe government’s indexation formula, the specification of this equation takes into accountthe sensitivity of the minimum wage to consumer price inflation and to wages. However,we made some changes compared to the government formula: we used variables for pricesand wages directly available from the model as proxies for variables actually used by thegovernment (consumer price index excluding tobacco of the first income quintile and manualworkers basic hourly wage); we included long run anchors in the formula for ensuring thelong run stability of the ratio of the minimum wage with respect to the average wage, whilethis long run stability is ensured by the government through additional positive exogenousshocks referred to as "coups de pouce".

Figure 4.5.1: Growth rate of the real efficient minimum wage

-.02

-.01

.00

.01

.02

.03

.04

.05

.06

-.02

-.01

.00

.01

.02

.03

.04

.05

.06

1998 2000 2002 2004 2006 2008 2010 2012 2014 2016

Growth rate of the real efficient minimum wage

YOY growth rate of the real efficient minimum wage

Solving forward equation (49), ignoring the explosive solution and accounting for the

46

balanced growth path, we obtain the wage Phillips curve that we take to the data:

πW,t = β0 + [∆et + πt] + β1(πC,t−1 − πt−1) + β2[πW,t−1 −∆et−1 − πt−1 − β1(πC,t−2 − πt−2)]

+β3

(∆4(wmt−1 − et−1)− πt−1

)+ β4PV (u)t−1|t−2 + εwt

(52)According to equation (52) presented above, wage inflation is not really on the balanced

growth path (BGP) because β0 6= 0. We had to include it in the specification order to centerthe residuals which were negative on average. We used only one long run trend (πt) forall inflation types, including wage inflation. This trend was computed using a professionalforecast of consumer price inflation, see subsection 3.1.2. The mean of the latter was higherthan the mean of any other inflation over the estimation sample period (1997Q2 - 2017Q4),implying that the wage inflation trend [∆et+ πt] was on average higher than the mean of thisseries during this period, which leads to negative residuals. In FR-BDF we modify equation(52) so that the constant disappears with time and πwt converges toward the balanced growthpath.

Note that the present value of the expected sum of future unemployment gaps PV (u)t−1|t−2

enters the equation in t− 1 and hence is based on the information of t− 2. In order to com-pute this variable, we added an auxiliary equation to the core of E-SAT that is in line withOkun’s law: {

ut = β0yt−1 + ut

ut = ρut−1 + εt⇒ ut = β0(yt−1 − ρyt−2) + ρut−1 + εt (53)

The estimates of the policy function, together with those of the auxiliary equation, areavailable in Table 4.5.2. It is interesting to mention that parameter β0 from equation (53) isvery close to the Okun’s law coefficient of the model which was evaluated to be 1/3 and wascomputed as an elasticity of the unemployment gap with respect to the output gap after ashock to government spending.

Table 4.5.2: Policy function of the expected discounted sum of the future unemploymentgaps

VAR Model variables Policy function Auxiliary equationPV (u)t|t−1 ut

yt−1 -0.02 -0.27 [0.06]it−1 − it−1 0.11 -πQ,t−1 − πQ,t−1 -0.01 -yEA,t−1 -0.01 -πEA,t−1 − πEA,t−1 0.01 -yt−2 - −0.27· (-0.94)ut−1 0.25 0.94 [0.036]

The estimates of equation (52) are presented in Table 4.5.3. The wage Phillips curvestrongly influences the relation between the unemployment rate and inflation of the model.

47

Using estimated parameters, we compute the Phillips slope(∂πt∂ut

)in partial equilibrium

making the following assumptions:

• to account for price indexation, we assume that a 1pp change in πW,t results in a 1 ppchange in πC,t.

• we set the Okun’s law parameter to 3 (which is consistent with an estimate of theauxiliary equation), i.e. a 1pp increase in the output gap leads to a 1/3pp decrease inthe unemployment gap.

∂πt∂ut

=−0.32 · [3 · (−0.02)− 0.25]

[(1− 0.24) ∗ (1− 0.32)− 0.22]= 0.34 (54)

This leads to an elasticity of inflation with respect to the output gap of 0.45 in annual terms,which is close to the estimate of 0.3 obtained by Chatelais et al. (2015).39

Table 4.5.3: Coefficients and standard errors of the wage Philips curve equation

Coef. s.e.

β0 −5 · 10−4 4 · 10−4

β1 0.24 0.1β2 0.32 0.1β3 0.22 0.1β4 -0.32 0.2

R2 = 0.18

Dynamic contributions Figure 4.5.2 shows how various components contribute dynam-ically to the variation in wage inflation. Price and wage inflation trends play a major rolein describing the dynamics of πW,t. Expectations also have a significant impact on the wageinflation. They were one of the main forces that drove wage inflation downward after thesovereign debt crisis of 2011. Conversely, the efficient minimum wage pushed wages upwardduring the period 2004-2014. This upward pressure came partly from the upward harmo-nization of the various minimum hourly wages, which were brought about by the reductionin working time.

39To estimate ∂πt

∂ytthey use quarterly data from the euro area over the period 1997Q1-2014Q4. Note that

here the output gap is the difference between observed GDP and potential GDP.

48

Figure 4.5.2: Dynamic contributions, wage inflation

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0.0

0.5

1.0

1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

1998 2000 2002 2004 2006 2008 2010 2012 2014 2016

Consumption inflation Nominal efficient minimum wage

Unemployment expectations Long-run anchor of inflation

Efficiency trend Residual

Wage inflation

49

4.5.2 Labor demand

Table 4.5.4: Variables used in section 4.5.2

Notation Description

nt Total employment of market branches, thou-sands of persons, in log

n∗S,t Target salaried employment of market branches,thousands of persons, in log

nS,t Salaried employment of market branches, thou-sands of persons, in log

nNS,t Non-salaried employment of market branches,thousands of persons, in log

qt Value added of market branches, volume, in logqt Market branches output gappQ,t Value added price of market branches, in logwt Total labor cost per capita, gross wages plus em-

ployers’ social contributions, value, in loght Hours per workers, in loget Trend labor efficiency, in log

E-SAT See Table 4.2.1n∗t HP-trend target salaried employment, in logn∗t Gap of target salaried employment relative to

n∗t , in log

We estimate the equation for the employment of market branches based on employeesnS,t rather than total employment nt. While both time series are strongly correlated, aspecification based on employees performs better than one on total employment.40

Table 4.5.5: Coefficient and standard errors for the salaried employment target equation

Coefficient Estimate s.e.

b0 8.85e-2 0.07e-2

R2 = 0.98

40Specifically, when estimating a simple ECM specification augmented by a balanced growth neutralityterm (growth rate of filtered employment), an equation based on total employment is better describedby a reduced form autoregressive process, while the cointegrating relationship plays a insignificant role.Conversely, an equation based on salaried employment displays a significant cointegrating relationship, whichmotivates our choice.

50

Target As detailed in the presentation of the supply-block of FR-BDF, the labor demandequation is derived from the firms’ first-order condition for labor which equals marginalproductivity to marginal cost of labor, augmented by a mark-up. Hence, the target level ofsalaried employment n∗S,t is defined by

n∗S,t = b0 + qt − et − ht − σ(wt − pQ,t − et − ht) (55)

The target level of salaried employment depends on value added, trend efficiency andthe hourly efficient real wage (deflated by value added price pQ,t) with long run elasticityσ = 0.53 estimated first step with the business investment equation (see section 4.3 fordetails).41 The only estimated parameter is the intercept b0, which is shown in Table 4.5.5.

Short run equation The short run dynamics of salaried employment are described by afourth-order PAC equation augmented by the change of the market-branches value added gap∆qt, to account for an additional Okun’s effect in the short run, plus a growth neutrality term,to ensure that salaried employment grows at the same rate as the trend of the employmenttarget in the long run.

The expectation of the present-value change of employment target is decomposed intotwo components. First, we define the HP-trend of the salaried employment target n∗t . Sec-ond, we split the expectation into two components: the trend growth rate in the salariedemployment PV (∆n∗S)t|t−1 and the change in the salaried employment gap PV (∆n∗S)t|t−1.The decomposition of expectations is intended to capture labor hoarding effects in the data.For instance, in the event of a negative transitory shock to the employment target n∗S,t, em-ployers would expect the target to return to its long run trend n∗S,t and would not reducetheir demand for labor as much as in a model without expectations. Finally, ω representsthe share of the non-stationary components of the expected changes in the target and is re-quired to ensure growth-neutrality in the long run (see section 3.2 for details). The salariedemployment equation (56) was estimated from 1997Q1 to 2017Q4. Results are shown inTable 4.5.6.

∆nS,t = β0

(n∗S,t−1 − nS,t−1

)(56)

+PV (∆n∗S)t|t−1 + PV (∆n∗S)t|t−1

+β1∆nS,t−1 + β2∆nS,t−2 + β3∆nS,t−3

+(1− β1 − β2 − β3 − ω)∆n∗S,t+β4∆qt + εt

41We use total labor cost wt from national accounts in the estimation of the target employment equation.The French government introduced the CICE tax credit to decrease cost of labor on medium and lowwages, which does appear in national accounts’ production subsidies and not in labor cost. In unconditionalsimulations and forecasting, we instead use the CICE-adjusted labor cost. Since total labor cost enters boththe employment and VA price equations, two of the three key equations of FR-BDF’s supply block, weconfirmed that the CICE-adjustment of total labor cost has no significant impact on our estimates; theseresults are available upon request.

51

Table 4.5.6: Coefficients and standard errors for the salaried employment short run equa-tion

Coefficient Estimate s.e.

β0 0.06 0.02β1 0.87 0.11β2 -0.30 0.15β3 0.17 0.10β4 0.15 0.03ω 0.26 -

R2 = 0.92

Expectations We construct the expected present value change in the salaried employmentgap PV (∆n∗S)t|t−1 by adding an AR(1) auxiliary equation for n∗t with links to E-SAT corevariables for the French economy. The auxiliary equation is:

n∗S,t = β0yt−1 + β1(it−1 − it−1) + β2(πQ,t−1 − πQ,t−1) + β3n∗S,t−1 + εt (57)

The estimated coefficients of equation (57) and the policy function for PV (∆n∗S)t|t−1 areshown in Table 4.5.7. The salaried employment gap depends significantly on its lag withcoefficient 0.67 and the output gap with coefficient 0.30, which is consistent with the orderof magnitude of Okun’s Law (see the estimates of the wage Phillips curve auxiliary equationin Table 4.5.2 for example). Conversely, it does not significantly depend on the short-terminterest rate and the French VA price inflation gaps but we chose to keep them in theauxiliary equation. Finally, the policy function associated with PV (∆n∗S)t|t−1 depends onall core variables of E-SAT plus the employment gap n∗S,t−1. As mentioned earlier, our

Table 4.5.7: Coefficients of policy function and auxiliary equation for expectation of thechange in the target employment gap

VAR model Policy function Auxiliary equationPV (∆n∗S)t|t−1 n∗S,t

yt−1 0.02 0.30 [0.09]it−1 − it−1 -0.03 0.07 [0.3]πQ,t−1 − πQ,t−1 0.02 0.16 [0.13]yEA,t−1 0.01πEA,t−1 − πEA,t−1 0.00n∗S,t−1 -0.05 0.67 [0.09]

Note: standard errors are in brackets. R2 = 0.82 for the auxiliary equation.

decomposition of expectations is intended to capture "labor hoarding" effects. In practice,it materializes as a negative coefficient associated with n∗S,t−1. In the case of a negative

52

transitory shock on n∗S,t−1, it would result in a positive effect on PV (∆n∗S)t|t−1 which woulddampen the negative effect on employment.

Finally, the expected present value trend growth rate of the salaried employment targetPV (∆n∗S)t|t−1 is constructed from a calibrated unit-root process for ∆n∗S,t and does notdepend on any E-SAT core variables. This choice is motivated by the I(2) nature of HPfiltered trends. As a result, the policy function for this expectation is reduced to:

PV (∆n∗S)t|t−1 = ω∆n∗S,t−1 (58)

Dynamic contributions We compute dynamic contributions for equation (56) and showthe results in Figure 4.5.3. We observe that movements in target employment account for alarge share of the variance of employment, given some delay due to strong persistence in theshort run equation. In turn, the expected change in the target employment gap and expectedtrend growth of target employment significantly dampen the target’s contribution, whichwe interpret as a "labor hoarding" effect. The equation also displays a significant Okun’slaw effect (through the output gap in E-SAT and in the short run equation) which helpsto explain employment dynamics, particularly during the crisis and the recovery. Positivecontributions of residuals in 2016 and 2017 mostly reflect the effect of CICE on employment,since we did not adjust total labor cost w for the CICE tax credit at the estimation stageand consequently in these dynamic contributions.42

4.6 Demand block

The demand block is composed of four main components: household consumption and in-vestment, business investment and external trade.

4.6.1 Household consumption

In FR-BDF the primary long run driver of household consumption Ct is permanent income,which together with a real interest rate gap determines the consumption target. Permanentincome is determined via an expectation term on the ratio of disposable income to outputmultiplied by the long run output. The interest rate gap is between the households’ realbank lending rate and the difference between the long run anchor of the real short rate.

The short run dynamics of consumption are determined with a first order PAC equationaugmented with terms representing the behavior of rule-of-thumb consumers (output gap),a wealth effect (change in the interest rate gap) and an indicator that is equal to 1 in thefirst quarter of 2011, -1 in the second and zero otherwise.43 The present value of the target issplit into expectations regarding the components of the target, which includes an expectationregarding permanent income, i.e. an expectation of an expectation.44

42As a result, dynamic contributions calculated with a CICE-adjusted total labor cost would displaysmaller residuals for the same period.

43The dummy is the result of a particular government policy in effect at the time, referred to colloquiallyas "prime à la casse" in French. The purpose of the policy was to subsidize the purchase of new vehicles,like the "cash for clunkers" policy implemented in the United States in 2009.

44Note that, as we do not have rational expectations, the expectation of an expectation does not equalthe expectation.

53

Figure 4.5.3: Dynamic contributions (in pp of growth rate), salaried employment

-1.2

-0.8

-0.4

0.0

0.4

0.8

-1.2

-0.8

-0.4

0.0

0.4

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2004 2006 2008 2010 2012 2014 2016

Target of employment

Expected trend growth of target employment

Expected change of target employment gap

Trend employment growth

Output gap

Residuals

Employment growth

Target The target for households’ consumption c∗t is based on a permanent income term –as described in (60) – represented by a transformation of a standard expectation on the ratioof real disposable household income yH,t to real long run GDP yt.45 Following Campbell &Mankiw (1989), we derive a log-linear consumption equation from the Euler equation andbudget constraint of a representative household.46

c∗t = α0 + PV (yH)t|t−1 + α1 (rLH,t − (it − πt)) (59)

The equation for the target, (59), also has an additional term that relates consumption toa real interest rate gap which is an attempt to capture long run effects of interest rates onconsumption. The two coefficients of the equation – the constant term α0 and the sensi-tivity to the interest rate α1 – are estimated as -0.16 and -0.95, respectively. The impliedintertemporal elasticity of substitution is approximately 0.1.

PV (yH)t|t−1 = PV (yH − y)t|t−1 + yt (60)

45Note that, contrary to FRB/US, the target of consumption depends here on aggregate permanent incomewith a single propensity-to-consume. In fact, when we tried to distinguish different types of income (wages,assets and transfers) with differentiated propensities, we obtained nonsensical estimates of these propensities.Another difference with respect to FRB/US is the absence of financial and housing wealth from the equation,this variable being insignificant in our estimation on French data.

46A detailed note is available on request.

54

Table 4.6.1: Variables used in section 4.6.1

Notation Description

ct Household consumption, volume (in log)yH,t Household disposable income, volume (in log)rLH,t Real household bank lending raterLH Steady-state of real household bank lending rateyt Gross domestic product, volume (in log)yt Long run trend of the volume of gross domestic product (in log)

δprime "Prime" dummy of 2011

E-SAT See Table 4.2.1∆weff,t Growth rate of real efficient wageut Unemployment gap

Note : the steady-state real household bank lending rate is defined by a spread over theshort-term real interest rate and is equal to (i− π) + sLH where sLH is the term premium.

Note that the construction of the permanent income term PV (yH)t|t−1 is slightly differentto the construction of the other expectations in FR-BDF. In particular, we assume that, dueto risk aversion and income uncertainty, the discount factor applied, i.e. the parameter β inthe equation is somewhat smaller than in the other cases at roughly 0.95. See Reifschneider(1996) for more details on the derivation of this discount factor. The core of the argumentrests on the fact that when optimal consumption – solved from a standard household problem– is related to expected uncertain income, this income stream has to be discounted usingnot just the real rate of interest as in the perfect foresight case, but also a risk adjustmentfactor that depends on household risk aversion and the variance of the income stream. Asexplained in section 6.3, this choice might play a role in the absence of forward guidancepuzzle in FR-BDF.

Short run equation The short run dynamics of household consumption are describedby a first order PAC equation augmented with several additional terms intended to capturephenomena related to deviations from the permanent income hypothesis. Equation (62)presents this equation and Table 4.6.2 the associated estimated coefficients. The augmentingterms include a term representing non-optimizing behavior by rule-of-thumb households(∆yt) and a term intended to capture interest rate effects via the change of the interest rategap (∆iLH,t − (∆it −∆πt)). The third ad hoc term is a dummy (δprime) equal to 1 in the firstquarter of 2011, -1 in the second quarter of 2011 and 0 otherwise which reflects a particulargovernment policy (car scrapping allowance), which increased the demand for cars in early2011. It is included only to improve the empirical fit.

The standard formulation of expectations with respect to changes in the target for con-sumption has been replaced here with expectation terms on the changes in the variouscomponents of the target itself, including an expectation on permanent income, i.e. an ex-pectation of an expectation, which we denote with PV2 (yH − y)t|t−1. The stationarity term

55

of the equation has been modified due to this change: as the expectation terms on gaps arealready stationary, the term representing the non-stationary component of expectations isnow unnecessary and can be omitted.

∆ct = β0

(c∗t−1 − ct−1

)+ β1∆ct−1 (61)

+PV2 (yH − y)t|t−1

+α1

(PV (rLH)t|t−1 −

(PV (i)t|t−1 − PV (π)t|t−1

))+ (1− β1) (∆(yt)−∆ (yH,t − yt))+β2 (∆yt) + β3 (∆rLH,t − (∆it −∆πt))

+β4δprime

Table 4.6.2: Coefficients and standard errors of the short run equation for householdconsumption

Coefficient Estimate s.e.

β0 0.12 0.05β1 -0.08 0.09β2 0.26 0.11β3 -0.71 0.45β4 0.007 0.002R2 = 0.54

Expectations There are a total of five expectation terms that appear in the various equa-tions of the household consumption block. The most important of these describes expec-tations regarding the ratio of real disposable income to real output, and is used to modelpermanent income. The coefficients of the associated policy function and auxiliary equationsare presented in Table 4.6.3.

Note that the policy function for permanent income depends on the growth rate of thereal efficient wage ∆weff,t−1 and the unemployment gap ut−1. These variables were addedin order to be able to account for the direct effects of developments in the labor market onpermanent income. Note also that the intercept term in the policy function is assumed tobe a constant only in the historical sample; in simulation mode, it is assumed to adjust viaa learning rule to a level that is consistent with the steady state ratio of income to long runoutput.

Table 4.6.4 presents the policy function for expected permanent income. Note thatthe corresponding auxiliary equation is in fact given by the policy function described inTable 4.6.3. This policy function is also dependent on the growth rate of wages and theunemployment gap.

The final expectation terms are described in Table 4.6.6, which presents the policy func-tion and auxiliary equation for the household bank lending rate iLH,t, and Table 4.6.7, which

56

Table 4.6.3: Coefficients of the policy function and auxiliary equation for the expectation ofthe income-output ratio and auxiliary equations for the real efficient wage and unemploymentgap

VAR model Policy function Auxiliary equationsPV (yH − y)t|t−1 yH,t − yt ∆weff,t ut

Constant -0.29 −0.49 (1− 0.92) [0.009]yt−1 -0.036 0.95 [0.04]it−1 − it−1 -0.085πt−1 − πt−1 0.013yEA,t−1 0.007πEA,t−1 − πEA,t−1 -0.004yH,t−1 − yt−1 0.39 0.92 [0.036]∆weff,t−1 0.3 0.32 [0.18] 0.58 [0.00]ut−1 -0.21 -0.08 [0.06] 0.95 [0.04]yt - -0.25 [0.06]Note: standard errors in brackets. R2 = 0.91 for the auxiliary equation of yH,t − ytR2 = 0.32 for ∆weff,t−1 and R2 = 0.92 for ut−1

Table 4.6.4: Coefficients of the policy function for the expectation of permanent income

CoefficientVAR model PV2 (yH − y)t|t−1

Constant -0.043yt−1 -0.052it−1 − it−1 -0.012πt−1 − πt−1 0.002yEA,t−1 0.001πEA,t−1 − πEA,t−1 -0.001yH,t−1 − yt−1 0.034PV (yH,t−1 − yt−1) -0.12∆weff,t−1 0.029ut−1 -0.03

57

presents the policy functions for PV (i)t|t−1 and PV (π)t|t−1. The auxiliary equations for thetwo latter terms are simply the E-SAT core equations as described in Section 3.1.1, implyingthat appropriately defined policy functions depend only on the two variables themselves.

Note that the policy function of PV (iLH)t|t−1 is defined in a non-standard way. Morespecifically, in order to ensure that the process has a zero mean and that the R2 of theregression is equal to 1, the two interest rates and inflation are centered and the long runanchors of the short rate and inflation are separated from their contemporaneous observationsinto new explanatory variables. The long-run anchor of the lagged real bank lending rate(rLH) is set equal to the steady state of the real short-run rate (i−π) augmented by a specificspread defined below (sLH).

The structure of the auxiliary equation for iLH,t is described by (62) and the estimatedcoefficients are presented in Table 4.6.5. The steady-state term premium sLH = 1.12%(annualized) is the sum of the steady-state premium of the bank lending rate over the 10-year government rate in equation (68) and the steady-state term premium component of the10 year government rate in equation (96). It can be loosely described as the total premiumover the 3 month Euribor.

rLH,t = β0rLH,t−1 + (1− β0) (it−1 − πt−1 + sLH) + β1 (it−1 − it−1) + β2 (πt−1 − πt−1) (62)

Table 4.6.5: Coefficients and standard errors of the auxiliary equation for the householdreal bank lending rate

Coefficient Estimate s.e

β0 0.88 0.02β1 0.12 0.02]β2 -0.06 0.02R2 = 0.98

Table 4.6.6: Coefficients of the policy function for the expectation of the household realbank lending rate

CoefficientVAR model PV (iLH)t|t−1

yt−1 -0.002it−1 − i 0.039it−1 − i 0.012πt−1 − π -0.001πt−1 − π 0.033yEA,t−1 0.001πEA,t−1 − πEA,t−1 0.005rLH,t−1 − rLH -0.056

58

Table 4.6.7: Coefficients of policy functions for expectations of it and πt

Policy function Policy functionVAR model PV

(∆i)t|t−1

PV (∆π)t|t−1

it−1 − i -0.011 -πt−1 − π - -0.036Note: Auxiliary equation defined in E-SAT core equations

Dynamic contributions Figure 4.6.1 describes how the various terms of the short runequation of the consumption block contributed to variation in the growth rate of consump-tion. They mostly contributed by permanent income. Variation in the long run output alsoplayed a role, albeit much smaller; during the financial crisis, however, it had a rather largenegative effect. The other terms have individually relatively unimportant effects. Finally,the significant effect of the "prime" dummy, alternating in sign, can be seen quite clearly inthe first and second quarters of 2011. Note that the contributions of the terms iLH,t, it andπt have been grouped into a single term "interest rate gap", as have the contributions of thecorresponding expectations. As shown by these contributions, interest rate changes play inthe end a small role in dynamics of French consumption.

59

Figure 4.6.1: Dynamic contributions, household consumption, in pp of growth rate

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

1998 2000 2002 2004 2006 2008 2010 2012 2014 2016

Consumption target Expectation of interest rate gap

Change in interest rate gap Expected change of income-output ratio

Change in target for income-output ratio Change in long run output

Change in output gap Prime dummy

Residual Consumption growth

60

4.6.2 Business investment

Table 4.6.8: Variables used in section 4.6.2

Notation Description

IB,t Investment, volume, NFCs, FCs and sole proprietorsrKB,t Real user cost of capital for firmswacct Weighted average cost of capitalδt Depreciation of firm capital stock

PCI,t Deflator, business investment goodPQ,t Deflator, value addedqt Value added, market branches, in log

E-SAT See Table 4.2.1

What we call here business investment groups investment of different types of firms(non-financial and financial corporations, as well as sole proprietors).47 In FR-BDF businessinvestment IB,t is strongly based on economic theory. The target is derived from a first ordercondition for capital for a firm with a CES production function. The short run dynamicsof business investment are assumed to follow a second order PAC equation, with an ad hocterm representing a business cycle-based demand for firms that do not fully optimize. Thestandard expectation term in the short run equation is split into the components of thetarget and defined in terms of gaps from trends in order to obtain a dampening of partialequilibrium impulse responses to shocks that affect expectations.

Target The target of firms’ investment I∗B,t is derived from a standard profit maximizationproblem for a competitive firm that has a CES production function and no investmentadjustment costs. The elasticity of the target I∗B,t to the real user cost of capital rKB,t isrepresented by the parameter σ that is estimated to be 0.53. See section 4.6.2 for detailsregarding the construction of the user cost. The other coefficient in the equation – α0 – isestimated to be 0.016. Finally, I∗/K∗ is the optimal investment-capital ratio and is assumedto be equal to the historical mean.

log I∗B,t = α0 + qt − σ log rKB,t−1 + logI∗

K∗(63)

Short run equation The short run dynamics of investment are determined by a secondorder PAC equation. The estimated coefficients are presented in Table 4.6.9. The expec-tation term of the equation has been replaced by expectations regarding gap terms of thecomponents of the target – qt and rKB,t – where the gap is measured with respect to thelong run trends of these variables. As a consequence of this choice, the stationarity term

47In particular, investment in social housing is included in this variable, as this type of investment isachieved in France by non-financial companies.

61

of the equation has been modified. In particular the term representing the non-stationarycomponent of expectations is now unnecessary, as the expectation terms on gaps are al-ready stationary. The last term in the equation is an ad hoc term that represents businesscycle-based demand intended to represent the nonoptimizing behavior of some firms.

∆ log IB,t = β0 log

(I∗B,t−1

IB,t−1

)+ β1∆ log IB,t−1 + β2∆ log IB,t−2 (64)

+PV (∆q)t|t−1 − σPV (∆ log rKB)t|t−1

+ (1− β1 − β2) (∆(qt−1)− σ∆ log(rKB,t−1))

+β3 (∆qt−1 −∆(qt−1))

Table 4.6.9: Coefficients and standard errors of the short run equation for business invest-ment

Coefficient Estimate s.e.

β0 0.085 0.029β1 0.29 0.14β2 0.2 0.1β3 0.58 0.36

R2 = 0.52

User cost of capital for firms The real user cost of capital for firms is determined viaequation (65), which relates the user cost to a Weighted Average Cost of Capital (WACC),depreciation δt and the ratio of the price of investment goods and the price of value added.48

The nominal rate is deflated with the term PV (πQ)t|t−1, i.e. expected value added inflation.Note that there is an implicit assumption of no adjustment costs to investment in the speci-fication of the equation; such costs would imply that the price of investment is not equal tothe price of the capital stock, i.e. Tobin’s Q is not equal to unity. In addition, in this casea more theoretically rigorous specification would replace value added inflation with inflationof the price of capital. Note that PV (πQ)t|t−1 is determined with E-SAT; the associatedpolicy function is described in Table 4.6.10. The discount rate used in this computation is(

11+i

)0.25

≈ 0.998 where i ≈ 2.29% (annualized) is the sample mean of it over the full sample.

rKB,t =[wacct + δt − PV (πQ)t|t−1

] PIB,tPQ,t

(65)

48In this formula, we do not take into account the effect of the capital income tax on the user cost ofcapital, as is done in Bardaji et al. (2017). This topic is left for further research.

62

Table 4.6.10: Coefficients of the policy function for expected inflation

CoefficientVAR model PV (πQ)t|t−1

Constant 0.002Yt−1 0.04it−1 − it−1 -0.18πt−1 0.097πt−1 0.45yEA,t−1 0.019πEA,t−1 − πEA,t−1 -0.004

Expectations The standard expectation term describing behavior of the investment targetthat appears in the short run equation has been split into the time-varying components ofthe target: the value added of the business sector and the user cost of capital for businessinvestment. More specifically, the expectations are formed on gaps of these terms withrespect to their trend processes.

The coefficients of the policy functions and the corresponding auxiliary equations forthese two terms are presented in Tables 4.6.11 and 4.6.12. The policy function for thegap of the user cost of capital includes the non-standard term it−2 − it−2 which has beenimplemented with the auxiliary variable xt = it−1. Note that none of the test statisticsassociated with these estimates have a meaningful interpretation as the dependent variables(the expectations) were created using the independent variables.

Table 4.6.11: Coefficients of the policy function and auxiliary equation for the expectationof market value added

Policy functionVAR model PV (∆q)t|t−1 Auxiliary equation

yt−1 0.035it−1 − it−1 -0.101πt−1 − πt−1 0.027yEA,t−1 0.015πEA,t−1 − πEA,t−1 -0.002rKB,t−1 0qt−1 -0.071 0.59 [0.048]yt - 0.61 [0.067]Note: standard errors in brackets. R2 = 0.90 for the auxiliary equation

Dynamic contributions Figure 4.6.2 describes how the various components contributedynamically to the variation in business investment. The main driver of variation is thegrowth of business output, i.e. investment is significantly driven by the business cycle and the

63

Table 4.6.12: Coefficients of policy function and auxiliary equation for expectation of usercost of capital

Policy functionVAR model PV (∆ log rKB)t|t−1 Auxiliary equation

yt−1 0it−1 − it−1 0.24 4.45 [2.36]it−2 − it−2 -0.13πt−1 − πt−1 0yEA,t−1 0.012πEA,t−1 − πEA,t−1 0.038rKB,t−1 -0.055qt−1 0Note: standard errors in brackets. R2 = 0.63 for the auxiliary equation

demand that firms are facing. The user cost of capital is also responsible for a notable shareof movements in investment, in particular with a negative contribution during the financialcrisis. Of the expectation terms, the one related to the user cost is the most important: itsimpact can be seen especially in 2009, where it has a strong positive contribution. Firmsexpect the positive shocks to the user cost to be temporary and thus reduce their investmentto a lesser extent, i.e. the expectation term has a dampening effect on responses to shocks.

64

Figure 4.6.2: Dynamic contributions, business investment, in pp of growth rate

-8

-6

-4

-2

0

2

4

-8

-6

-4

-2

0

2

4

1998 2000 2002 2004 2006 2008 2010 2012 2014 2016

User cost of capital Business output growth

Expected business output growth Expected user cost of capital

Growth of potential output Trend growth of user cost of capital

Residual Investment growth

65

4.6.3 Household investment

Table 4.6.13: Variables used in section 4.6.3

Notation Description

IH,t Household investment, volumeyH,t Household disposable income, volume (in log)iLH,t Real household bank lending rateyt Gross domestic product, volume (in log)yt Long run trend of the volume of gross domestic product (in log)pIH,t Deflator, new housing investment (in log)pSH,t Deflator, existing housing stock (in log)pC,t Deflator, household consumption (in log)i10,t Yield on 10-year French government bonds

E-SAT See Table 4.2.1

Target The household’s target for investment follows (66). The associated estimationresults can be found in Table 4.6.14. The desired level of investment by households isassumed to depend on the permanent income term PV (yH)t|t−1 and the prices of new (pIH,t)and existing (pSH,t) housing relative to the price of the consumption good pC,t. The finalterm represents the real user cost of housing investment

(iLH,t − PV (πQ)t|t−2 + δH

). δH

represents the households’ depreciation rate and is calibrated at 1.8% per year.

log I∗H,t = log γ0 + PV (yH)t|t−1 (66)+γ1 (pIH,t−1 − pC,t−1) + γ2 (pSH,t−2 − pC,t−2)

+γ3 log(iLH,t−2 − PV (πQ)t−2|t−3 + δH

)Table 4.6.14: Coefficients and standard errors of the long run equation for householdinvestment

Coefficient Estimate s.e.

γ0 0.005 0.001γ1 -2.2 0.11γ2 0.55 0.036γ3 -0.071 0.023

R2 = 0.82

66

Short run equation The short run dynamics of household investment are described by(67), while Table 4.6.15 presents the relevant estimation results. We assume that m = 2,i.e. implying a single lag of household investment on the right hand side of the equation.Expectations regarding changes in the target have been split into gap and trend components.As the trend component is present in the specification, the standard PAC specification forensuring growth neutrality is applied, with ω, the share of the nonstationary componentin expectations present in the equation.49 The equation also contains an ad hoc term, thecontemporaneous change in the output gap, to account for liquidity contrained householdsand direct effects of demand and the contemporaneous change in the deflator of existinghousing, centered by its trend, to account for short-term dynamics of housing prices.

∆ log IH,t = β0 log

(I∗H,t−1

IH,t−1

)+ β1∆ log IH,t−1 (67)

+PV(

∆ log I∗H

)t|t−1− PV

(∆ log I∗H

)t|t−1

+ (1− β1 − ω) ∆ log I∗H,t+β2∆yt + β3 (∆pSH,t −∆pSH,t)

Table 4.6.15: Coefficients and standard errors of the short run equation for householdinvestment

Coefficient Estimate s.e.

β0 0.056 0.019β1 0.62 0.069β2 0.34 0.20β3 0.32 0.09ω 0.36 -

R2 = 0.87

Expectations Expectations regarding changes in the target for household investment havebeen split into two different components: the trend of the target and a gap term measuringdeviations from this trend. The estimation results for the policy functions and auxiliaryequations relating to the second term can be found in Table 4.6.16. The policy function forthe change in the trend component is a simple AR(1) with a coefficient equal to 0.25, theshare of the nonstationary component in expectations.

Bank lending rate for households The dynamics of iLH,t are described by (68). Changein the bank lending rate is related to the difference between the rate and the 10-year gov-ernment rate via an error correction mechanism, with α0 representing a long run premium

49See section 3.2.1 for details.

67

Table 4.6.16: Coefficients of policy functions and auxiliary equations for expectations ofthe gap of the household investment target

Policy function Auxiliary equationVAR model PV

(∆ log I∗H

)t|t−1

yt−1 0.029 0.38 [0.26]it−1 − it−1 -0.15 -0.89 [0.96]πt−1 − πt−1 0.035 0.49 [0.54]yEA,t−1 0.004 -πEA,t−1 − πEA,t−1 -0.012 -log I∗H,t−1 -0.044 0.71 [0.096]Note: standard errors in brackets. R2 = 0.62 for the auxiliary equation

over the public rate.

∆iLH,t = α1 (iLH,t−1 − i10,t−1 − α0) + α2∆i10,t + α3∆i10,t−1 + α4∆iLH,t−1 + α5∆i10,t−2 (68)

Table 4.6.17: Coefficients and standard errors of the equation for household bank lendingrate

Coefficient Estimate s.e.

α0 0.001 0.0005α1 -0.047 0.022α2 0.075 0.031α3 0.26 0.038α4 0.66 0.077α5 -0.19 0.071

R2 = 0.52

Price of housing stock The dynamics of the price of the existing housing stock are givenby (69), which is a relatively simple AR (2) process in the change of the logarithm of theprice level with a constrained intercept.

∆pSH,t = ρ0∆pSH,t−1 + ρ1∆pSH,t−2 + (1− ρ0 − ρ1) pit (69)

Dynamic contributions Figure 4.6.3 shows how household investment is mostly drivenby the variation in the target, which follows the business cycle with a delay. Expectations, inparticular the gap component, also have a notable role, but in the opposite direction vis-à-vis

68

Table 4.6.18: Coefficients and standard errors of the equation for the price of housing stock

Coefficient Estimate s.e.

ρ0 0.48 0.1ρ1 0.43 0.1

R2 = 0.72

the business cycle and household income, reflecting a dampening effect. Finally, the housingprice gap and, to a lesser extent, the ad hoc output gap term have an amplifying effect onhousehold investment, e.g. driving it further down during the Great Recession and pusingit further up during the immediate recovery.

Figure 4.6.3: Dynamic contributions, household investment, in pp of growth rate

-6

-5

-4

-3

-2

-1

0

1

2

3

-6

-5

-4

-3

-2

-1

0

1

2

3

2006 2008 2010 2012 2014 2016

Household investment target Expected change in target, gap component

Expected change in target, trend component Change in trend of target

Output gap Housing price gap

Residuals Household investment

4.6.4 External trade

As we show below, we were able to obtain high estimates for the price elasticities in equationsof exports and non-energy imports excluding energy, thanks to the inclusion of the weight ofemerging countries in the first case and a goods variety indicator in the second. As explainedin section 5.1, these elasticities play a key role in the long run convergence of the model.

Exports The real export equation is estimated as a one-step ECM. The estimation periodis 2000Q2-2017Q4.

69

Table 4.6.19: Variables used in section 4.6.4

Notation Description

Xt Exports (volume)WSt World supply for France (volume)PX,t Export pricePMO,t Import price, other than energyPCX,t Foreign competitors price’ (export side)ωt Weight of emerging countries (in log)∆q Long run anchor of the output growth rateMO,t Non-energy importsQSM,t Value added, market sectorMNRJ,t Energy imports (volume)PMNRJ,t Energy import pricePQ,t Value added priceTt Time-varying trend

DMNRJ,t Import intensity-adjusted measure of aggregatedemand (IAD) for energy imports

DMO,t Import intensity-adjusted measure of aggregatedemand (IAD) for imports other than energy

Ct Household consumption, volumeIB,t Investment, volume, NFC, FC and sole proprietorIH,t Household investment, volumeCG,t Government consumption, volumeIG,t Government investment, volume

70

Target World demand is the only regressor in the equation that has a non-zero growthrate.50 Hence, we need a unit coefficient in front of the latter in order to satisfy the balancedgrowth path condition. The ratio PX,t

Pcx,tin the target is used as a price competitiveness

indicator specific to exports. The weight of emerging countries is also used here in some senseto reveal the competitiveness of French producers not captured by the price difference.51

x∗t = β0 + dW,t + β1(pX,t − pCX,t) + β2ωt (70)

For notations, see Table 4.6.19. Estimated coefficients are shown in Table 4.6.20.

Table 4.6.20: Estimates and calibrated parameters, exports

Long run Short runCoef. s.e. Coef. s.e.

β0 11.93 0.02 0.83 0.05β1 -1.27 0.41 -0.15 0.06β2 -0.63 0.07 - -

R2 = 0.69 (1 step estimation)

Short run equation

∆xt = β0∆dW,t−1 + (1− β0)∆q + β1

[xt−1 − x∗t−1

]+ εt (71)

Dynamic contributions Variation in the world demand for French goods is the maincontributor to French export fluctuations, see Figure 4.6.4. The weight of emerging countriesin world trade helps explaining the underperformance of French market shares from 2003 to2014.

Imports We model separately volumes for energy and other imports. The choice to splitthe total import volume and price is due to the heterogeneity in coefficients of adjustment (forprice) and elasticity of substitutions (for volumes). Total imports are then simply modeledthrough a chained price aggregation.

Import intensity-adjusted measures of aggregate demand (IAD) for non-energy importsand energy play an important role in the equations of imports. They are computed withweights based on input-output tables including the option to re-export imported goods orservices, see equations (72) and (73).

50 By world demand we mean the weighted sum of imports by trade partners. For more details aboutcomputational aspects of these variables see Hubrich & Karlsson (2010).

51 This variable captures the fact that a share of foreign demand is not "addressed" to France but ismet by emerging economies whose competitiveness is not adequately measured by our price competitivenessindicator, possibly because these countries expand at the extensive margin (new varieties, new destinations)rather than at the intensive margin (increased price competitiveness on existing markets and products).

71

Figure 4.6.4: Dynamic contributions, Export (in pp of growth rate)

-8

-6

-4

-2

0

2

4

6

-8

-6

-4

-2

0

2

4

6

02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17

Log of world demand for France Log of price ratio (export price over competitors prices)

Log of weight of emerging countries Residuals

Log of real export

DMO,t = 0.194Ct + 0.094CG,t + 0.252IB,t + 0.197IG,t + 0.150IH,t + 0.305Xt (72)

DMNRJ,t = 0.037Ct + 0.012CG,t + 0.013IB,t + 0.014IG,t + 0.016IH,t + 0.026Xt (73)

Non-energy imports This equation is estimated as a two step ECM. The estimationsample is 2001Q2-2014Q4.

Targetm∗O,t = β0 + dMO,t + β1 (pX,t − pMO,t) + β2 (wst − qN,t) (74)

The variable wst − qN,t is a proxy for the relative variety of goods in the world withrespect to France, i.e. the variety of foreign goods relative to the variety of French supply.The former is proxied by the weighted average of exports of countries supplying France(weights are their share in French imports). The construction is symmetrical to that ofworld demand for French goods (see footnote 50). The latter is proxied by the Frenchlong run value added of market branches. It helps in attributing a substantial role to pricecompetitiveness. Price competitiveness is proxied here by the relative price of exports withrespect to the price of non-energy imports, as we consider the price of exports as a betterproxy of the price of tradable goods produced locally than the value added price.

72

Short run equation

∆mO,t = β0∆dMO,t−1 + (1− β0)∆y + β1

[mO,t−1 −m∗O,t−1

]+ εt (75)

For notations, see Table 4.6.19. Estimated coefficients are shown in Table 4.6.21.

Table 4.6.21: Estimates and calibrated parameters, non-energy imports

Long run Short runCoef. s.e. Coef. s.e.

β0 2.79 0.3 1.91 0.13β1 1.11 0.19 -0.19 0.1β2 0.39 0.04 - -

R2 = 0.98 R2 = 0.67

Dynamic contributions The main driver of the non-energy import dynamics is the im-port intensity-adjusted measure of aggregate demand for such imports, see Figure 4.6.5.

Figure 4.6.5: Dynamic contributions, non-energy imports (in pp of growth rate)

-6

-4

-2

0

2

4

6

-6

-4

-2

0

2

4

6

2002 2004 2006 2008 2010 2012 2014 2016

IAD for imports other than energy Relative price (exports/imports other than energy)

World demand/VA Residual

Imports other than energy

73

Energy imports (volume) This equation is estimated as a one-step ECM, i.e. target andshort run equations are estimated simultaneously. The estimation sample is 2000Q1-2017Q4.

Targetm∗NRJ,t = dMNRJ,t + β0 (pMNRJ,t − pQ,t) + β1 + β2Tt (76)

Short run equation

∆mNRJ,t = (1− β0 − β1)∆y + β0∆dMNRJ,t + β1∆mNRJ,t−1 + β2

[mNRJ,t−1 −m∗NRJ,t−1

]+ εt

(77)In order to recover a meaningful coefficient β0 of the price ratio in the target equation

(76), we have to estimate both equations in one step without any long run restrictions,i.e. allowing the constant in the short run equation to be freely estimated, which yieldsβ0 = −0.16, see Table 4.6.22. In the next step we calibrate β0 to −0.16 and estimate otherparameters of (77).

Table 4.6.22: Estimates and calibrated parameters, energy imports (in pp of growth rate)

Long run Short runCoef. s.e. Coef. s.e.

β0 -0.16 0.08 1.17 1.15β1 0.35 0.12 -0.36 0.11β2 -0.004 0.002 -0.12 0.08

R2 = 0.26 (One-step estimation)Note: β0 was estimated separately. See text for details.

Dynamic contributions Since the R2 of equation (77) is very small indicating a poorempirical fit of this equation, most of the dynamics of the growth rate of energy importscome from residuals. Plotting dynamic contributions is in this case unnecessary.

74

4.7 Demand deflators

We model eight deflators with econometric equations: household consumption and invest-ment deflators, government consumption deflator, corporate investment deflator, export de-flator, import excluding energy and energy deflators, total import deflator and value addeddeflator of market branches.52 For all, except the last two, we use common transverse princi-pals. First, their long run relation is modeled as a linear combination of market value addedand import prices. The market value added deflator is described in section 4.4. The importdeflator is computed as an accounting identity: value of imports over volume of imports.The latter is defined using annual prices of imports excluding energy and annual prices ofenergy. Second, the weight of the import price is defined using weights of the import contentof demand (henceforth, IAD weights). These weights are calculated using input-output ta-bles including a re-exporting option of imported goods or services. Third, the deflators aremodeled as error correction equations.

In the case of the household consumption deflator, the corporate investment deflator andthe household investment deflator, we model their homologies excluding VAT, which arelinked to the former through a tax rate which is exogenous. For example, let the householdconsumption deflator including and excluding VAT be respectively πC+vat,t and πC,t, and thetax rate – τC , then their relation is simply presented as:

PC+vat,t = (1 + τC)PC,t (78)

4.7.1 Household consumption deflator excluding VAT

Table 4.7.1: Variables used in section 4.7.1

Notation Description

πC,t Consumer price inflation excluding VATp∗C,t Target of consumer price (in log)pC,t Consumer price (in log)pQ,t Value added price of market branches (in log)pM,t Import price (total) (in log)

PMNRJ,t Energy import pricePt Exogenous trend priceπt Long run anchor of inflationTt Time-varying trend

The household consumption deflator plays a significant role in FR-BDF. Householdsuse it to index their gross wage and the government uses it to index the minimum wage, seeequation (52). It is also used to deflate disposable income and many items of public spendingare indexed on it. Its equation is estimated as an ECM in two steps on the 2000Q1 - 2017Q4sample.

52Sometimes in the text we refer to the value added price (deflator) of market branches simply as the(market) value added price (deflator).

75

Target In order to center the residuals we had to include a time-varying trend (Tt) whichwould become a constant at some point so that the deflator converges to the balanced growthpath. Parameter β0 is calibrated to be consistent with the corresponding IAD weight, i.e.with the share of imports in consumption. For notations see Table 4.7.1. Estimated andcalibrated parameters are in Table 4.7.2.

p∗C,t = (1− β0)pQ,t + β0pM,t + β1Tt + β2 (79)

Table 4.7.2: Estimates and calibrated parameters, household consumption deflator

Long run Short runCoef. s.e. Coef. s.e.

β0 0.23 - 0.63 0.07β1 5.4 · 10−4 0.3 · 10−4 0.16 0.07β2 -0.043 0.002 0.05 0.003β3 - - -0.05 0.03

R2 = 0.82 R2 = 0.85

Short run equation In the short run, in order to obtain a non-linear effect of oil priceshocks, we use the absolute variation in the energy import price – normalized by the exoge-nous trend price Pt to preserve the BGP of FR-BDF – rather than the growth rate of theenergy import price.53

πC,t = (1− β0 − β1)πQ,t + β0πQ,t + β1πQ,t−1 + β2∆(PMNRJ,t/Pt)

+β3

[pC,t−1 − p∗C,t−1

]+ εt

(80)

Dynamic contributions See Figure 4.7.1. The value added price played an importantrole mainly before 2008. After that, the dynamics of consumption inflation were mainlydriven by the energy import price from the short run equation.

4.7.2 Business investment deflator

The business investment deflator πIB,t influences the model through the real cost of capital.It is estimated as a one-step ECM: the target and the short run equations are estimatedsimultaneously.

Target The parameter β0 is calibrated to be consistent with IAD weights, i.e. with theshare of imports in business investment.. For notations see Table 4.7.3. Estimated andcalibrated parameters are in Table 4.7.4. The estimation period is 1994Q4-2017Q4.

53This exogenous trend price is normalized to 1 at the base year 2014 and with a constant growth rateequal to FR-BDF’s steady-state inflation rate π.

76

Figure 4.7.1: Dynamic contributions, consumer price inflation (in pp of growth rate)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2002 2004 2006 2008 2010 2012 2014 2016

Long-run anchor of inflation Market value added price

Import price Energy import price

Time-varying constant Residuals

Consumption inflation excluding VAT

Table 4.7.3: Variables used in section 4.7.2

Notation Description

πIB,t Business investment price inflationp∗IB,t Target of business investment price (in log)pIB,t Business investment price (in log)pQ,t Value added price of market branches (in log)pMO,t Import price (excluding energy)(in log)pMNRJ,t Energy import price (in log)πQ,t Long run trend of the value added price inflationπM,t Import price inflation (total)

p∗IB,t = (1− β0)p∗Q,t + β0 [(1− β1)pMO,t + β1pMNRJ,t] (81)

Short run equation The short run equation is estimated as an ECM. The estimationperiod is 1995Q1 - 2017Q4.

πIB,t = (1− β0 − β1)πQ,t + β0πM,t + β1πIB,t−1 + β2

[pIB,t−1 − p∗IB,t−1

]+ εt (82)

77

Table 4.7.4: Estimates and calibrated parameters, business investment deflator excludingVAT

Long run Short runCoef. s.e. Coef. s.e.

β0 0.27 - 0.11 0.03β1 0.25 0.01 0.42 0.1β2 - - -0.06 0.03

R2 = 0.38 (1 step estimation)

Dynamic contributions The import price played an important role in explaining thehikes in the deflator from 2004 to 2008, and from 2010 to 2012, ensuring a good econometricfit, see Figure 4.7.2. Note that the import price here also includes energy and non-energyimport prices from the short run equation.

Figure 4.7.2: Dynamic contributions, business investment deflator excluding VAT (in ppof growth rate)

-0.8

-0.4

0.0

0.4

0.8

1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

2002 2004 2006 2008 2010 2012 2014 2016

Long-run anchor of inflation Value added price

Import price Residual

Growth rate of business investment deflator excluding VAT

4.7.3 Household investment deflator

This deflator, πIH,t, is estimated as a two-step ECM. The estimation period is 1995Q1-2017Q4.

78

Table 4.7.5: Variables used in section 4.7.3

Notation Description

πIH,t Household investment deflatorp∗IH,t Target of household investment price (in log)pIH,t Household investment price (in log)pQ,t Value added price of market branches (in log)pM,t Import price (excluding energy) (in log)pimmo,t Price of existing housing stock (in log)δ99−08,t Dummy 1: 1 during 1999-2008, 0 otherwiseδ99Q4,t Dummy 2: 1 during 1999Q4, 0 otherwiseδ08Q4,t Dummy 3: 1 during 2008Q4, 0 otherwiseT08Q3,t Time-varying trend from 1995Q1, a constant after 2008Q3.πQ,t Long run trend of the value added price inflation

πimmo,t Growth rate of price of existing housing stock

Target The target of this deflator is a weighted average of the value added price and thetotal import price. The parameter β1 is calibrated to be consistent with IAD weights. Fornotations, see Table 4.7.5. In order to center the residuals a time-varying trend T08q3,t isintroduced. After 2008Q3 T08q3,t is equal to the last value of the series. Estimated andcalibrated parameters are in Table 4.7.6.

p∗IH,t = β0 + (1− β1)pQ,t + β1pM,t + β2T08q3,t + β3δ99−08,t (83)

Table 4.7.6: Estimates and calibrated parameters, household investment deflator

Long run Short runCoef. s.e. Coef. s.e.

β0 -0.35 0.005 0.9 0.18β1 0.17 - 0.005 0.001β2 0.006 0.0001 -0.03 0.007β3 -0.06 0.004 -0.03 0.006β4 - - -0.09 0.04

R2 = 0.99 R2 = 0.48

Short run equation

πIH,t = (1− β0)πQ,t + β0πQ,t + β1T08q3,t + β2δ99Q4,t + β3δ08Q4,t

+β4

[pIH,t−1 − p∗IH,t−1

]+ εt

(84)

Dynamic contributions See Figure 4.7.3.

79

Figure 4.7.3: Dynamic contributions, household investment deflator excluding VAT (in ppof growth rate)

-4

-3

-2

-1

0

1

2

3

-4

-3

-2

-1

0

1

2

3

02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17

Long-run anchor of inflation Value added price

Import price Time-varying trend

Dummy (2008Q4) Dummy (1999Q4)

Dummy (1999Q3-2007Q4) Residual

Growth rate of household investment deflator excluding VAT

4.7.4 Export deflator

The growth rate of the export deflator, πX,t is estimated as a one-step ECM. The estimationperiod is 2000Q1-2017Q4.

Target The parameter β0 is calibrated to be consistent with IAD weights, i.e. with theimport content of exports. For notations, see Table 4.7.7. Estimated and calibrated param-eters are in Table 4.7.8. Foreign competitors’ export price pcx,t is a weighted sum of tradepartners’ export prices.54

p∗X,t = (1− β2) [(1− β0)pQ,t + (β0 − β1)pMNRJ,t + β1pMO,t] + β2pCX,t (85)

Short run equation

πX,t = (1− β0 − β1)πQ,t + β0πQ,t + β1πM,t + β2

[pX,t−1 − p∗X,t

]+ εt (86)

Dynamic contributions The dynamics of the export price are mainly driven by theimport price, see Figure 4.7.4.

54For more details about the computational aspects of this variables see Hubrich & Karlsson (2010).

80

Table 4.7.7: Variables used in section 4.7.4

Notation Description

πX,t Export price inflation (in log)p∗X,t Target of export price (in log)pX,t Export price (in log)pQ,t Value added price of market branches (in log)pMO,t Import price (excluding energy) (in log)pMNRJ,t Energy import price (in log)pCX,t Foreign competitors’ price (export) (in log)πQ,t Long run trend of the value added price inflationπM,t Import price inflation (total)

Table 4.7.8: Estimates and calibrated parameters

Long run Short runCoef. s.e. Coef. s.e.

β0 0.33 - 0.89 0.12β1 0.28 0.02 0.48 0.03β2 0.27 0.07 -0.1 0.03

R2 = 0.86 (One-step estimation)

4.7.5 Import deflators

We separately model the growth rate of the non-energy import price and the growth rateof the energy import price. These annual prices are then used to aggregate total imports involume across both components. Having defined total imports in value as a simple sum ofthe two components, we compute the total import price as an accounting identity:

PM,t =PMO,tMO,t + PMNRJ,tMNRJ,t

Mt

(87)

The decision to split the total import volume and price was motivated by the heterogene-ity in the adjustment coefficient (for price) and the elasticity of substitution (for volume).

Non-energy import price The non-energy import price is estimated with a two-stepECM on the 2000Q1 - 2017Q4 sample.

Long run equation

p∗MO,t = β0pQ,t + (1− β0)pCM,t + β1ωt + β2 (88)

For notations see Table 4.7.9.

81

Figure 4.7.4: Dynamic contributions, export price (in pp of growth rate)

-3

-2

-1

0

1

2

-3

-2

-1

0

1

2

2002 2004 2006 2008 2010 2012 2014 2016

Long-run anchor of inflation Value added price

Import price Competitors export price

Residual Growth rate of export deflator

Short run equation

πMO,t = (1− β0 − β1)πt + β0πCM,t + β1πMO,t−1

+β2

[pMO,t−1 − p∗MO,t

]+ β3 + εt

(89)

Estimation results are presented in Table 4.7.10. Foreign competitors’ price on importside pcm,t is a weighted sum of export prices with weights computed on import trade part-ners.55

Dynamic contributions The dynamics of the non-energy import price are mainly drivenby the competitors’ price (import), see Figure 4.7.5. An increasing weight of emergingcountries is pushes PMO,t−1 down over the whole sample period, while added value price hasan upward effect on it.

Energy import price The energy import price is estimated with a one-step ECM on the2000Q1 - 2017Q4 sample.

Long run equation

p∗MNRJ,t = (1− β0)pQ,t + β0(pOIL,t − usdt) + β1 (90)

For notations see Table 4.7.9.55For more details about computational aspects of this variables see Hubrich & Karlsson (2010).

82

Table 4.7.9: Variables used in section 4.7.5

Notation Description

p∗MO,t Target of non-energy import price (in log)pMO,t Non-energy import price (in log)pQ,t Value added price (in log)

p∗MNRJ,t Target of energy import price (in log)pMNRJ,t Energy import price (in log)pOIL,t Oil price (Brent, in log)pCM,t Foreign competitors’ price (import) (in log)πMO,t Import price inflation (excluding energy)πQ,t Long run trend of the value added price inflationπM,t Import price inflation (total)ωt Weights of emerging countries (in log)usdt Dollar/euro exchange rate (in log)

Table 4.7.10: Estimates and calibrated parameters, non-energy import price

Long run Short runCoef. s.e. Coef. s.e.

β0 0.56 0.04 0.21 0.03β1 -0.31 0.01 0.36 0.08β2 0.06 0.002 -0.07 0.03β3 - - -0.002 0.0005

R2 = 0.54 R2 = 0.64

Short run equation

πMNRJ,t = (1− β0)πt + β0(∆pOIL,t −∆usdt) + β1

[pMNRJ,t−1 − p∗MNRJ,t

]+ εt (91)

Results are presented in Table 4.7.11.

Dynamic contributions The dynamics of the energy import price are mainly driven bythe euro-denominated oil price, see Figure 4.7.6.

4.7.6 Government consumption deflator

The main explanatory variable of the dynamics of the government consumption deflator(PG,t) is the public sector salary. We have chosen to calibrate the elasticity of this deflatorwith respect to the efficient wage (πW,t−1 −∆et−1) to be equal to the share of public sectorwages (in value) in government spending (in value) in 2014Q4. This is estimated to be 0.54,see equation (92):

πG,t = 0.54(πW,t−1 −∆et−1) + (1− 0.54)πt−1 + εt (92)

83

Figure 4.7.5: Dynamic contributions, non-energy import price (in pp of growth rate)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

03 04 05 06 07 08 09 10 11 12 13 14 15 16 17

Long-run anchor of inflation Value added price

Competitors import price Weights of emerging countries

Residuals Growth rate of import price excluding energy

4.8 Financial block

This section deals with the financial block: we cover, successively, the short and long gov-ernment rates, the exchange rate and the determination of how the financial asset positionsand asset income of the various sectors.

4.8.1 Short-term interest rate

The short rate it is measured by the 3-month Euribor. In simulation its dynamics aredetermined by a Taylor rule reacting to euro area inflation and the output gap, given by(93). Since in typical simulation applications these two variables are assumed exogenous,this equation in practice reduces to a simple AR (1). In forecasting the short rate is fullyexogenous.

(1− λiL) (it − ıt) = (1− λi) (αi (πea,t−1 − πea,t−1) + βiyea,t−1) (93)

Note that (93) is simply the E-SAT core equation for the short rate; details relating to itsestimation can be found in section 3.1.1.

Variation in the short rate is the primary driver of financial variables’ dynamics in FR-BDF. Even though the short rate itself does not appear in any of the main behavioralequations, it determines the dynamics of the 10-year rate, which in turn either affects thereal sector directly or is a key determinant of other interest rates (e.g. the user cost of capitalfor firms). Furthermore, as the short rate is a core component of the E-SAT expectations

84

Table 4.7.11: Estimates and calibrated parameters

Long run Short runCoef. s.e. Coef. s.e.

β0 0.82 0.01 0.55 0.02β1 -3.5 0.05 -0.65 0.06

R2 = 0.92 (1 step estimation)

model, it has an effect on agents’ behavior via all the expectation terms in the backward-looking setup.

As in the E-SAT model, the long-run anchor of the short-term interest rate, measured bythe 5-year ahead forward rate of the 3-month Euribor rate, is simply modeled as an AR(1)process: (

1− λiL)ıt = εi,t (94)

While we choose in E-SAT to calibrate the persistence of this variable to a value λi = 0.985in order to get term premia more in line with those assessed by models which take intoaccount the possibility of zero lower bound, we do not impose such an assumption in theFR-BDF model. We set the persistence to its estimate equal to λi = 0.91. Such an estimateis obtained using as a sample the pre-crisis period and it implies a fast normalization ofinterest rates in unconditional simulations.56

4.8.2 Long-term government interest rate

The long rate is measured by the return on 10-year French government bonds. Its dynamicsare determined by the term structure equation (95) which relates the 10-year rate to anexpectation component PV (i)t|t−1 and the term spread s10,t, which we assume to follow asimple AR(1), as per (96). The estimation results are presented in Table 4.8.2. PV (i)t|t−1 isdetermined using E-SAT; the relevant policy function is described in Table 4.8.3.57 Figure4.8.1 plots the elements of equation (95).

i10,t = PV (i)t|t−1 + s10,t (95)

s10,t = (1− ρ10) s10 + ρ10s10,t−1 (96)The term structure equation (95) has its theoretical foundations in an approximation wherethe bond is modeled as having an infinite maturity with coupon payments that decay ata geometric rate. The calibration of the decay is chosen so that the distance between ourapproximated bond and the 10 year bond is minimized. The implied theoretical equationfor i10,t, the yield at maturity of a hypothetical long term bond, is then

i10,t = (1− κ10)∞∑s=0

κs10 (it+s) + s10,t (97)

56In conditional forecasts, this choice has no influence, as this variable is simply extrapolated using technicalassumptions of the forecast related to 5-year ahead futures.

57Note that no additional auxiliary equation is needed as it is included in the E-SAT core.

85

Table 4.8.1: Variables used in section 4.8

Notation Description

it 3-month Euribori10,t Yield on 10-year French government bondswacct Weighted average cost of capitaliCOE,t Firm cost of equityiLB,t Bank lending rate for firmsiBBB,t Yield on French BBB rated corporate bondss10,t Term spread between 10-year and 3-month French government bondsYFj,t Net property income for agents of type jiFj,t Rate of return on financial assets for agents of type jτTj,t Financial transfers paid by agents of type j to householdsWj,t Net stock of financial assets for agents of type jτTj,t Financial transfers paid by agents of type j to the householdsWj,t Net stock of financial assets for agents of type jBj,t Net financing need for agents of type jYt Long run real outputPY ,t Price of long run outputYIG,t Net interest rate income, governmentYIGP,t Interest rate income paid, governmentYIGR,t Interest rate income received, governmentYOG,t Net other financial income, governmentYOGP,t Other financial income paid, governmentYOGR,t Other financial income received, government

86

Figure 4.7.6: Dynamic contributions, Energy import price (in pp of growth rate)

-40

-30

-20

-10

0

10

20

-40

-30

-20

-10

0

10

20

2002 2004 2006 2008 2010 2012 2014 2016

Long-run anchor of inflation Value added price

Oil price (in euro) Residuals

Growth rate of energy price

where κ10 is the decay – linked to the duration of the bond – and s10,t is the theoretical termspread.

Table 4.8.2: Coefficients and standard errors, term structure equation

Coef. Estimate s.e.

ρ10 0.80 0.06s10 0.15e-2 0.04e-2

The long rate plays an important role in FR-BDF as the foundation on which the con-struction of the rates paid by the private sector are based, particularly the user cost of capitalrKB,t and the household bank lending rate iLB,t. Furthermore, it is used to determine theincome received by the various agents on their financial assets.

4.8.3 Private interest rates

Weighted average cost of capital (WACC) The weighted average cost of capital wacct– a key component of the user cost of capital – is determined as a weighted average of themain components of funding used by French firms: Cost of Equity (COE) iCOE,t, the firmbank lending rate iLB,t and a bond rate iBBB,t which is represented by the rate on BBB-rated corporate bonds, as computed by Merrill Lynch-Bank of America. The weights on the

87

Figure 4.8.1: Long-term government interest rate and its components, annualized percent

-1

0

1

2

3

4

5

6

7

-1

0

1

2

3

4

5

6

7

1996 2000 2004 2008 2012 2016

Long-term government rate Term premiumExpectation component of long-term government rate

Table 4.8.3: Coefficients of the policy function for the expectation of the 3-month bondrate

VAR model Coefficient

Constant 0.003yt−1 0it−1 0.16i 0.51πt−1 − π 0yEA,t−1 0.026πEA,t−1 − πEA,t−1 0.041

components have been computed as historical averages of their shares in the liabilities ofFrench corporations. See Carluccio et al. (2019) for details. Figure 4.8.2 plots the WACCand its components.

wacct = 0.5iCOE,t + 0.3iLB,t + 0.2iBBB,t (98)

In simulation all the components of the WACC are determined as the sum of a spreadterm sj,t with j ∈ {COE,LB,BBB} and the long rate:

ij,t = sj,t + i10,t (99)

The three spreads are all assumed to follow simple AR(1) processes of the form (100) andare plotted in Figure 4.8.3. The corresponding estimated coefficients are reported in Table4.8.4.

88

Figure 4.8.2: Weighted average cost of capital and its components, annualized percent

0

2

4

6

8

10

12

0

2

4

6

8

10

12

1996 2000 2004 2008 2012 2016

Cost of Equity BBBBank lending rate, firms WACC

sj,t = sj (1− ρj) + ρjsj,t−1 (100)

Table 4.8.4: Coefficients and standard errors, spread equations

Coef. Estimate s.e.

ρCOE 0.92 0.05sCOE 1.40e-2 0.2e-2ρLB 0.77 0.05sLB 0.30e-2 0.05e-2ρBBB 0.94 0.026sBBB 0.02e-2 0.1e-2

Measure of cost of equity The observed series for iCOE,t used in the estimation have beencomputed using the methodology of Carluccio et al. (2019). The computation is based on anextension of the standard dividend-discount model in the style of Fuller & Hsia (1984), whichis used to compute the risk premium Rm,t for French firms. Combining this with separatelyestimated sensitivities to risk - i.e. Capital Asset Pricing Model (CAPM) betas - makes itpossible to construct observed iCOE,t.

89

Figure 4.8.3: Spreads between various rates and the 10 year government rate, annualizedpercent

-2

0

2

4

6

8

-2

0

2

4

6

8

1996 2000 2004 2008 2012 2016

Spread, cost of equity Spread, BBB bond rateSpread, bank lending rate

4.8.4 Exchange rates

There are two exchange rates in FR-BDF: the effective exchange rate of the euro area andthe exchange rate between the dollar and the euro.58 Both are expressed in direct quotes andmodeled using an uncovered interest rate parity (UIP) condition, which means that exchangerates are defined as functions of the countries’ interest rate differentials:

Ξt+1

Ξt

=(1 + iF,t)

(1 + it)(101)

where Ξt stands for the direct exchange rate (i.e. units of foreign currency per unit ofdomestic currency), it and iF,t are domestic and foreign short run interest rates, respectively.The UIP condition (101) requires that each asset must have the same nominal return andthis interdependence is ensured through capital movements between two countries such thatthe preceding no-arbitrage condition holds.

We define the log real exchange rate qt:

qt ≡ ξt + pEA,t − pF,t (102)

where pt and pFt are respectively the log of euro area and foreign prices, we can rewritethe UIP condition for the real exchange rate, in terms of interest rate and inflation ratedifferentials, in logs:

qt − (pEA,t − pF,t) = qt+1 − (pEA,t+1 − pF,t+1) + (it − iF,t)qt = qt+1 + (it − iF,t)− (πEA,t+1 − πF,t+1) (103)

58The effective exchange rate of the euro area measures the value of the euro with respect to a bundle of38 countries.

90

Solving forward equation (103) and using the definition of the log real exchange rate(102), we obtain equation (104) that we use as a benchmark for the empirical specificationof the two nominal exchange rates.

ξt + pEAt − pFt =∞∑κ=0

(it+κ − iF,t+κ

)−∞∑κ=0

(πEA,t+1+κ − πF,t+1+κ

)(104)

Table 4.8.5: Variables used in section 4.8.4

Notation Description

ξEA,t Effective exchange rate of the euro area, in logξ$,t Dollar/Euro exchange rate, in logit/iF,t Domestic/foreign short run interest rate

pEA,t/pF,t Euro area/foreign GDP deflator, in logπEA,t/πF,t Euro area/foreign quarterly inflation rate (GDP deflator

ξFR,X,t/ξFR,M,t French exchange rate on export/import sidePcm,t/Pcx,t Foreign competitors’ price (import/export)ωincm,t/ωincx,t Share of intra-EA trade flows in French imports and exportsP incm,t/P in

cx,t Intra-EA foreign competitors’ price (import/export)P excm,t/P ex

cx,t Extra-EA foreign competitors’ price (import/export)P excm,fc,t/P ex

cx,fc,t Extra-EA foreign competitors’ price expressed in foreign currency (import/export)

E-SAT variables See Table 4.2.1

Both nominal exchange rates ξEA,t and ξ$,t are estimated with a constant and an AR(1)process in residuals estimated on the sample 1999Q2-2010Q1 before the sovereign debt crisis.For notations see Table 4.8.5.

The infinite sums of the short run interest rates and inflation rates are computed asnon-discounted present values (PVnd) by inverting the corresponding models: the E-SATmodel in the case of it and πEA,t+1 and AR(1) models for iF,t and πF,t+1 (see equations(107) and (109)). The exchange rates of the euro vis-a-vis the dollar and the bundle of 38other currencies (ξ$,t and ξF,t respectively) are estimated in the following forms, first for theDollar/Euro exchange rate:

ξ$,t + pEAt − pFt = β +

[(PVnd(i)t|t−1 − PVnd(iF )t|t−1

)−(PVnd(πEA)t+1|t−1 − PVnd(πF )t+1|t−1

)]+ ηt

with (1− ρL)ηt = εt, which can be rewritten as follows:

(1− ρL)(ξ$,t + pEAt − pFt ) = (1− ρ)β + (1− ρL)

[(PVnd(i)t|t−1 − PVnd(iF )t|t−1

)−(PVnd(πEA)t+1|t−1 − PVnd(πF )t+1|t−1

)]+ εt (105)

91

And for the Euro effective exchange rate:

ξEA,t + pEAt − pFt = β +

[(PVnd(i)t|t−1 − PVnd(iF )t|t−1

)−(PVnd(πEA)t+1|t−1 − PVnd(πF )t+1|t−1

)]+ ηt

with (1− ρL)ηt = εt on, which can be rewritten as follows:

(1− ρL)(ξEA,t + pEAt − pFt ) = (1− ρ)β + (1− ρL)

[(PVnd(i)t|t−1 − PVnd(iF )t|t−1

)−(PVnd(πEA)t+1|t−1 − PVnd(πF )t+1|t−1

)]+ εt (106)

Estimated coefficients are presented in Table 4.8.6. Policy functions of the present valueof non-discounted sums of future domestic short run interest rates (PVnd(i)t|t−1) and of futuredomestic inflation rate (PVnd(πEA)t+1|t−1) are given in Table 4.8.7. It is interesting to notethat the semi-elasticity of PVnd(i)t|t−1 with respect to (it−1− i) is defined largely by the euroarea variables. If one considers the E-SAT model without euro area variables (leaving onlythree equations: French IS and Phillips curves and an AR(1) instead of Taylor rule) andinverts it, then this elasticity becomes 12.2 instead of 3.14.59

Table 4.8.6: Coefficients and standard errors of exchange rate equations

ξ$,t eq. 105 ξEA,t eq. 106 iF,t eq. 107 PVnd(iF ) eq. 108 πF,t eq. 109 PVnd(πF ) eq. 110Coef. s.e. Coef. s.e. Coef. Coef. Coef. s.e. Coef.

β 0.15 0.23 0.04 0.15 - - - - -γ - - - - - - - - -ρ 0.95 0.03 0.94 0.04 0.96 0.96 0.47 0.09 0.47

In order to construct PVnd(iF )t|t−1 we need a model for foreign short run interest rates.To proxy the latter, we apply the Federal Reserve’s 3-month interest rate modeled as anAR(1) process with mean reversion:60

iF,t = ρiF,t−1 + (1− ρ)i+ εt (107)

This implies the following present value of non-discounted sums of future foreign short runinterest rates:

PVnd(iF )t|t−1 =∞∑κ=0

(iF,t+κ − i) =ρ

(1− ρ)(iF,t−1 − i) (108)

For the sake of simplicity, ρ in equation (108) is set to be the same as that of the French shortrun interest rate.61 We apply the method to construct the PVnd(πF )t+1|t−1. We proxy the

5912.2 is obtained by computing λi

1−λi= 0.9243

(1−0.9243) , where λi is the persistence parameter of the short runinterest rate, see Table 3.1.1.

60This avoids modeling the US economy, which is assumed to be exogenous.61Such a simple relation as presented in (108) is not sufficient to recover the true value of this parameter.

92

foreign price level and inflation rate using the US GDP deflator and estimate the followingAR(1) process:

πF,t = ρπF,t−1 + (1− ρ)π + εt (109)

where π is the steady-state inflation rate.62 We finally obtain:

PVnd(πF )t+1|t−1 =∞∑κ=0

(πF,t+1+κ − π) =ρ

(1− ρ)(πF,t−1 − π) (110)

Table 4.8.7: Policy function of the expected sum of the future short run interest rate

VAR Model variables Policy function Policy functionPVnd(i) PVnd(πEA)

it−1 − i 3.14 -3.15it−1 − i 62.52 3.15QEA,t−1 1.26 0.40πEA,t−1 − π 1.64 0.12πEA,t−1 − π -1.64 13.11yt−1 - -πQ,t−1 − πQ,t−1 - -

Foreign competitors’ prices of exports and imports expressed in euros (Pcx,t and Pcm,trespectively) are modeled as weighted average of intra- and extra-EA foreign competitorsprices. Weights of competitors on the import side are computed as simple shares of intra andextra imports in French imports. On the export side, these weights are theselve computedwith weighted sums of export shares, in order to take into account thrird-market effects (seeHubrich & Karlsson 2010). We denote shares of intra-EA exports and intra-EA importsrespectively by ωinx,t and ωinm,t.63

Pcx,t = ωinx,t × P incx,t + (1− ωinx,t)× P ex

cx,t (111)

Pcm,t = ωinm,t × P incm,t + (1− ωinm,t)× P ex

cm,t (112)

Intra-EA foreign competitors prices for exports and imports (respectively P incx,t et P in

cm,t) areassumed exogenous, but further developments of FR-BDF may lead to link them to EuroArea variables.

Extra-EA foreign competitors prices for exports and imports in euros (respectively, P excx,t

et P excm,t) vary with the effective exchange rate of the euro area ΞEA and with foreign com-

petitors’ prices of exports and imports expressed in foreign currencies (respectively, Pcx,fc,tand Pcm,fc,t), which are exogenous:

P excx,t =

P excx,fc,t

ΞEA,t

(113)

62We therefore make the simplyfing assumption that the steady-state inflation rate is equal in the US andthe Euro Area.

63In the historical sample, the weights are observed at annual frequency and then interpolated at quarterlyfrequency. In simulation, we assume they are constant and prolong them at their last value.

93

P excm,t =

P excm,fc,t

ΞEA,t

(114)

Oil prices in euros are modeled as a product of oil prices in dollars and the dollar/euroexchange rate:

Poil,t =Poil,$,tΞ$,t

(115)

4.8.5 Net property income and net asset positions

The framework for determining the net property income YFj,t of the various agents j ∈{Firms, Government, Households, Non-profit organizations} of the model is based on

YFj,t = iFj,tWj,t−1 (116)

where iFj,t is the agent-specific rate of return on the net stock of financial wealth Wj,t−1 ofagent j.64 We denote the four agents with the subscripts F , G, H and N for brevity. Formost agents we deviate from this baseline. More specifically, we assume that for firms

YFF,t = iFF,tWF,t−1 − τTF,tYtPY ,t (117)

where the term τTF,tYt represents the real financial transfers made by the firms to householdsin order to stabilize their net asset ratio. Similarly, the financial income of the non-profitorganizations is modified to account for transfers τTN,tYt between them and households:

YFN,t = iFN,tWN,t−1 − τFN,tYtPY ,t (118)

The households’ financial income is then

YFH,t = iFH,tWH,t−1 + τTF,tYtPY ,t + τFN,tYtPY ,t (119)

i.e. they receive payments directly from their stock of wealth WH,t−1 and as transfers fromfirms and non-profit organizations. Note that Yt is long run real output and PY ,t is its price– the product is then nominal long run output.

The stocks of financial wealth Wj,t are assumed to evolve following

∆Wj,t = Bj,t (120)

where Bj,t is the net financing capacity of agents of type j, with the exception of firms, forwhom we also model the revaluations vF,t of their asset stock so that

∆WF,t = BF,t + vF,t (121)

vF,t = γYtPY ,t (122)

with γ = −0.018, which is the mean of the ratio of reevaluations to YtPY ,t.

64Data about net financial wealth comes from financial accounts.

94

The transfer policies of firms and non-profit organizations are given by

τTF,t = (1− ρstab,1) τTF,t−1 + ρstab,1τ∗TF (123)

−ρstab,2(−BF,t + γYtPYt

YtPY,t+WF

Y

exp (g + π)− 1

exp (g + π)

)

τTN,t = (1− ρstab,1) τTN,t−1 + ρstab,1τ∗TN − ρstab,2

(−BN,t

YtPY,t+WN

Y

exp (g + π)− 1

exp (g + π)

)(124)

where ρstab,1 = 0.1 and ρstab,2 = 0.1 are calibrated. τ ∗TF = 0.026 and τ ∗TN = 0.00026 areexogenous long run targets – constructed using a simulation method – for the rate at whichdividends and transfers are paid to households. WF

Y= −0.7 ∗ 4 and WN

Y= 0.02 ∗ 4 are

exogenous, calibrated targets for the ratio of assets to nominal output for firms and non-profit organizations, respectively. Note that these two rules are strongly based on a similarpolicy rule used by the government to ensure that its net financial asset-to-GDP ratio isstable in the long run:

τTG,t = (1− ρstab,1) τTG,t−1 + ρstab,1τ∗TG − ρstab,2

(−BG,t

YtPY,t+WG

Y

exp (g + π)− 1

exp (g + π)

)(125)

where WG

Y= −0.4 ∗ 4 and τ ∗TG = 0.16 are constructed similarly to their counterparts in

equations (123) and (124). The key difference is that this rule is not used to regulatefinancial transfers, but to determine the share of social transfers excluding unemploymentbenefits and social transfers in kind relative to nominal GDP. See section 4.10 for details.

The rates of return ij,t are assumed to follow for all agents j an autoregressive distributedlag process

ij,t = ρj,0 (1− ρj,1) + (1− ρj,1) i10,t + ρj,1ij,t−1 (126)

the coefficients of which are shown in Table 4.8.8. We assume that for all types of agentsρj,1 = 0.983 – calibrated so that the process has a half-life of 40 quarters – and that ρG,0 = 0to ensure convergence to i10,t. The long run premia are estimated from data as the historicalmeans of the ratio of total asset income to total asset stock minus the mean of i10,t.

Table 4.8.8: Coefficients and standard errors, asset return processes

Coef. Estimate

ρF,0 -0.0037ρG,0 0ρH,0 -0.0007ρN,0 -0.001

Finally, due to the requirements of accounting appropriately for government finances, wemodel separately the interest payments made by the public sector - YIGP,t - and asset incomepayments excluding interest income made by the public sector, i.e. YOGP,t, where "O" standsfor "other". First, the interest payments are computed as

YIGP,t = YIGR,t − YIG,t

95

i.e. as the difference between interest payments received YIGR,t and net interest income YIG,t.We assume that interest payments received YIGR,t are a constant share - 0.011% - of longrun output Yt, while the net interest income is computed as

YIG,t = YFG,t − YOG,t

i.e. as the difference between total net asset income YFG,t (described above) and net assetincome excluding interest payments YOG,t. This, in turn is simply the difference betweenreceipts and payments, i.e.

YOG,t = YOGR,t − YOGP,tboth of which are assumed to be constant shares of Yt - 0.006 and 0.05 percent, respectively.

4.9 Trends

FR-BDF includes several trend variables denoted by xt for any variable xt.65 These trendsare introduced for two main reasons. First, trend variables are often needed within thePAC framework to construct expectations for target variables x∗t . For instance, we quiteoften decompose expectations of target variables between the trend growth rate of the target(PV (∆x∗)t|t−1) and the change in the gap between the target and its trend (PV (∆x∗)t|t−1),as in section 4.5.2. Second, we need several trend variables to evaluate long run output. Forinstance, we use trends of labor force, hours worked per employee or non-market branchesGDP.

In the historical sample and for the purpose of estimation, these trend variables are ob-tained by applying the HP filter on quarterly data with λHP = 1600. Then, in conditionalforecasting and unconditional simulation, trend variables are projected using single exponen-tial smoothers. For any variable xt and its trend xt, we use the following single exponentialsmoother:

xt = ρHT (xt−1 + gx) + (1− ρHT )xt + εt (127)

where gx is the long run growth rate of the variable; for stationary variables, this termnaturally cancels out.

For some variables, we observed highly persistent residuals εt which had undesirableproperties in simulation when residuals are set to zero.66 Consequently, for these variables,we use a modified single exponential smoother with AR(1) residuals:

xt = ρHT (xt−1 + gx) + (1− ρHT )xt + ηt

ηt = ρTV Pηt−1 + εt

which we express using a nonlinear expression:

xt−ρTV P xt−1 = ρHT(xt−1−ρTV P xt−2 +(1−ρTV P )gx

)+(1−ρHT )

(xt−ρTV Pxt−1

)+εt (128)

65These filtered variables are referred to as "trend" variables as they capture low-frequency evolutions. Insome cases, these variables can, however,be stationary.

66In practice, variables for which we observe large and persistent residuals often appear to be nonstationary.

96

Both ρHT and ρTV P are calibrated in an ad hoc fashion. We set ρHT = 0.95 to achievearound 90% of convergence toward the observed level xt within 40 quarters and ρTV P = 0.83such that only 10% of the initial residuals persist at the end of the usual forecasting horizon(12 quarters).

Equations (127) and (128) are the general framework for modeling trends within FR-BDF. However, in practice, we can slightly deviate from these equations in order to preservethe model’s dynamic properties. As any smoother algorithm, the exponential smoother usesthe actual variable to evaluate its trend. In turn, in the PAC framework, the trend of thetarget and the gap between the target and its trend are included in expectation variables.This creates a loop between endogenous variables (e.g. the target of employment) andtheir trends, which can have undesirable properties (e.g. amplifying or creating oscillatorydynamics) in conditional forecasting and unconditional simulation exercizes.

From what precedes, we conclude that trends should be exogenous when feasible. Thus,we choose to anchor the exponential smoother not to the actual variable that we wishto smooth, but rather to the long run equilibrium of the variable. For instance, in theemployment equation presented in section 4.5.2, the trend of the employment target n∗S,t isanchored to the long run equilibrium level of salaried employment of market branches n∗S,t,defined by:

n∗S,t = log

((1− uN,t)νψtPOP t

)(129)

where ν is the historical average of the ratio of salaried employment to total employmentof market branches, ψt is the trend share of market branches employment in total employ-ment, POP t is the trend of the labor force and uN,t is the long run equilibrium rate ofunemployment.

Another example is the trend of the marginal return on capital services Q′K,t presented insection 4.3. In this case, we anchor the exponential smoother not to the observed marginalreturn but rather to the steady state of the capital demand equation (26). We define Q′K,tas follows:

Q′K,t ≡ µrK,tPQ,t

(130)

where rK,t is evaluated using the long run anchors of WACC, inflation expectations, a con-stant depreciation rate of capital services δ. In particular, we use the observed relative priceof investment with respect to the value added price of market branches, because we arenot able to obtain an analytical solution for it. As a result, Q′K,t is not fully but merelyquasi-exogenous.

Four trend variables have a particular status and depart from the modelling frameworkthat we have presented here: it the long run trend of the short run interest rate, πt the longrun trend of the value added price inflation of market branches, πEA,t the long run trendof euro area GDP deflator inflation and Et the trend labor efficiency. See section 3.1.2 fordetails about the first three variables. Finally, trend labor efficiency follows a deterministictrend, as described in section 4.3.

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4.10 Accounting framework and public finances

In our accounting framework we decompose production and labor market variables intomarket and non-market branches.67 As the accounting framework is based on quarterlynational accounts, some choices were made according to data availability in these accounts.For example, as the volume of value added is available only in branch accounts and not insector accounts, we include in the accounting framework branch accounts in value and inchained price volume, but only sector accounts in value.68 The branch account decompositionalso plays an important role in the interaction with the HICP model used for inflationforecasting, known as MAPI (Model for Analysis and Projection of Inflation in France, seede Charsonville et al., 2017, for a detailed presentation).

The second important decomposition is based on sector accounts. In this dimension, wedistinguish five economic agents:

• firms, which correspond to non-financial and financial corporations;

• government, which corresponds to general government;

• households, which correspond to households including unincorporated enterprises;

• non-profit organizations, which correspond to non profit institutions serving households(NPISH);

• rest of the world.

The model includes a detailed account in value for each agent (from its value added to itsborrowing capacity). For firms, we group together non-financial and financial corporations,because such a split is generally unnecessary for participating in the Eurosystem’s forecastingexercises (except for some credit variables).69 For households, we include unincorporatedenterprises, because the headline variables needed for forecasting, i.e. the savings ratio andthe gross disposable income, should concern households including those agents.70 Althoughtheir weight in GDP is very small, we include an account of non-profit organizations in orderto cover all sectors of national accounts.71 Finally, the bottom lines of the accounts of thegovernment and rest of the world determine two important headline variables, namely thegovernment balance and the current account. The variables of the rest of the world aregenerally simply determined by the variables of other agents through accounting identities.

Although this sector decomposition is generally used in FR-BDF, we make an exceptionfor business investment: as investment decisions of unincorporated enterprises are closer to

67As indicated earlier in the section about he supply block, market branches correspond to branches AZto MN in Insee codes, while non-market branches correspond to OQ.

68For chained price aggregation, as in quarterly national accounts provided by Insee or Eurostat, our modeluses annual overlap formulas.

69For details about the Eurosystem’s staff macroeconomic projection procedures, see the guide providedhere.

70Moreover, quarterly national accounts do not contain a full account of households excluding unincorpo-rated enterprises.

71Another possibility would have been to include this NPISH sector within the field of households but thiswould have not complied with the need to forecast the saving rate of households only.

98

those of corporate firms than to those of households, we estimate behavioral equations forbusiness investment including unincorporated enterprises on one side and household invest-ment excluding these unincorporated enterprises on the other. In addition, bridge equationsare used to ensure consistent dynamics of corporate firms’ and households’ investments withthese equations.

The government block is particularly detailed and designed in such a way as to allowinteractions with the public finance model of the Banque de France, known as MAPU (Ma-quette Agrégée des finances PUbliques). In particular, we have created two modes for manyvariables in this block:

• A forecasting mode, where many nominal variables are exogenized and come directlyfrom the public finance model;

• A simulation mode, where these variables are endogenous and depend on exogenouseffective tax rates or exogenous ratios.

More precisely, in the simulation mode, we adopt the following common principles onreceipt and spending sides:

• On the receipt side, each receipt is determined by an exogenous effective tax rate andon an endogenous tax basis;

• On the spending side, some spending variables are directly related to macroeconomicaggregates with effective rates. For example, unemployment benefits are directly re-lated to unemployment and wage per capita. For other spending variables, such asintermediate consumption or government investment, the ratio of their volume relativeto long run output is assumed to be exogenous.

In the simulation mode, one exception to these principles concerns social transfers excludingunemployment benefits and social transfers in kind TG,t. As is the case for many variableson the spending side, their volume is related to long run output through a ratio, τTG,t, butthis ratio is endogenized in this mode with equation (125) to ensure the convergence of thegovernment’s net asset ratio toward its long run target (see section 4.8.5).

Finally, we take into account two peculiarities of the government account compared toother sector accounts. First, for this agent, the operational surplus is not determined as itsvalue added net of paid compensations, but is assumed to be determined as an exogenousshare of long run output. Indeed, this surplus is mainly its consumption of fixed capital and,to simplify, we assume the share of this consumption in long run GDP to be stable. Once theoperating surplus is determined, the value added of government is simply determined as thesum of this surplus, paid compensations and other taxes on production (net of correspondingsubsidies). Second, another peculiarity of the government account concerns final consump-tion, which is an accounting construction: in fact, the government does not consume publicservices, while households and firms do; however, this consumption cannot be attributed tohouseholds or to firms since there are no observable monetary transactions; hence, nationalaccountants attribute it to the general government itself. For these reasons, governmentconsumption is determined as the sum of government value added, government intermediateconsumption and paid social benefits in kind, net of government sales of services.

99

4.11 Model changes for conditional projections within BMPE ex-ercises

In conditional forecasts, FR-BDF is modified to comply with the Eurosystem’s Broad Macroe-conomic Projection Exercises (BMPE) guidelines (ECB, 2016). Table 4.11.1 summarizesmodel changes for conditional forecasts.

First, some exogenous variables of FR-BDF – which are projected at constant growthrates (e.g. world demand, competitors’ prices) – are projected using the Eurosystem’s as-sumptions in conditional forecast exercises.

Second, several endogenous variables are taken from the Eurosystem’s assumptions de-termined within the BMPE, particularly variables in the financial block: both expected termstructure and Uncovered Interest Parity (UIP) equations no longer determine the long-termnominal rate, the nominal effective exchange rate and dollar/euro nominal exchange rate.As a result, crude oil prices in euro and competitors’ prices are also exogenized, followingthe Eurosystem’s assumptions.72 The Eurosystem’s assumptions are determined and revisedseveral times within an exercise. More importantly, these assumptions account for externalspillovers, using intermediate projections from National Central Banks (NCBs) within theEurosystem.

Third, three auxiliary models are used to compute (i) bank lending rates (BLR) to firmsand households, (ii) HICP forecasts, (iii) public finances variables. BLR models used withinthe Eurosystem NCBs must integrate common features but can also be partly country-specific. For instance, bank lending rates must be obtained by spread equations over long-term nominal rates and may integrate country-specific channels to account for financialfrictions. In contrast, models used for public finances and inflation forecasting are totallyspecific to the Banque de France.

Fourth, we use labor force projections from the French national institute of statistics andeconomic studies (Insee) and the long run equilibrium unemployment rate from an externalassessment described in section 4.3.3, which deals with the definition of long-run output andof its components.

HICP inflation forecasts are performed within MAPI (Model for Analysis and Projec-tion of Inflation in France, see de Charsonville et al., 2017, for a detailed presentation).MAPI conditional forecasts use FR-BDF’s macroeconomic projections as inputs. FR-BDFconditional forecasts then use updated inflation forecasts from MAPI to determine the con-sumption price deflator, which is in practice exogenized. Both models iterate several timeswithin an exercise to achieve convergence.

In the same spirit, public finances are projected outside FR-BDF, within a model knownas MAPU and developed by the Banque de France’s Public Finances Studies Division.MAPU conditional forecasts use FR-BDF projections as inputs and iterate with FR-BDFuntil convergence, in particular regarding the level of spending, revenues and deficit. Here,the difficulty lies in the discrepancy between public finance variables which are set on anannual basis, and quarterly national accounts which are the basis of FR-BDF. Public financevariables need to be adjusted at quarterly frequency, sometimes using external information

72In unconditional simulations, competitors’ prices are partly determined by effective exchange rates andby exogenous competitors’ prices in foreign currency. Crude oil prices’ in euro depend on the dollar/euronominal exchange rate.

100

about the timing of the implementation of fiscal measures. Finally, the minimum wage equa-tion (51) is modified in conditional forecasting exercises in order to comply with the effectiveminimum wage formula, in particular regarding the long run stability condition that we in-troduced to preserve the balanced-growth path of the model. In conditional forecasts, theminimum wage equation is replaced by the following equation:

wmt = δq1

(π4,C,t + 0.5 [π4,W,t − π4,C,t]

)+ εt (131)

which closely follows the minimum wage formula. Finally, additional judgments are necessaryin forecasting exercises, due to discrepancies between variables used in the model (consumerprice inflation, average wage per capita) and those used by the true formula (HCPI for thebottom 20% of households, blue-collar real wage per capita).

Table 4.11.1: Summary of model changes for conditional projections

Variables Status in FR-BDF Determination in conditional fore-casts

Euro-area output gap Endogenous Eurosystem assumptionsEuro-area GDP deflator Endogenous Eurosystem assumptionsWorld demand to French exporters(volume)

Exogenous Eurosystem assumptions

Brent price in dollars Exogenous Eurosystem assumptionsBrent price in euros Endogenous Eurosystem assumptionsExchange rates (dollar/euro, NEER) Endogenous Eurosystem assumptionsCompetitors’ prices for exports and im-ports in euros

Endogenous Eurosystem assumptions

Short-term interest rate (3-month Eu-ribor)

Exogenous Eurosystem assumptions

Expected short-term interest rate (5-year futures of 3-month Euribor)

Exogenous Eurosystem assumptions

Long-term interest rate (OAT 10 years) Endogenous Eurosystem assumptionsLong-term bank lending rate to firms Endogenous Auxiliary BLR model with Eurosystem

assumptionsLong-term bank lending rate to house-holds

Endogenous Auxiliary BLR model with Eurosystemassumptions

HICP: Total, Core, Energy and Food Endogenous Auxiliary model for inflation forecastsof the Banque de France (MAPI)

Public finance variables: VA sub-components, taxes and subventions, so-cial contributions and transfers at cur-rent prices

Endogenous Auxiliary model for Public financesforecasts of the Banque de France(MAPU)

Labor force Exogenous Insee projectionsLong run equilibrium unemploymentrate

Exogenous Banque de France projections

Note : the word "exogenous" is used here for variables that do not depend on other endogenous variablesof the model.

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5 Main model properties under VAR-based expectationsIn this section, we present the main model properties of the baseline version of the model,where expectations are assumed to be VAR-based.73 In the first subsection, we describe thelong run equilibrium and convergence mechanisms of FR-BDF. In the second subsection, wepresent impulse response functions (IRFs) for four demand shocks: a shock on the short runinterest rate, a term-premium shock on the long run interest rate, a foreign demand shock, agovernment spending shock, and three supply shocks: an oil price shock, a cost-push shockand a permanent labor efficiency (or labor-augmenting productivity) shock.

5.1 Long run convergence in unconditional simulations

In this subsection, we start by checking the convergence of unconditional simulations of FR-BDF toward a balanced growth path (BGP). To do so, we run an unconditional simulation ofthe model under VAR-based expectations from 2018Q1 to 2300Q1. All residuals are set at 0at the beginning of the simulation. Exogenous variables are extrapolated with the three keygrowth rates that prevail along the balanced growth path: ∆y for real variables homogenousto output, ∆e for real variables homogenous to labor productivity and π for nominal variableshomogenous to a price. Based on these simple assumptions on exogenous variables, the pointof this simulation is not to provide a realistic forecast along the transition, but to assess howfast the model converges towards the balanced growth path.

As shown in Figure 5.1.1, the output and inflation gaps – the inflation gap being definedas the gap between value added price inflation of market branches and its steady state levelπ = 1.9% – converge toward 0 in around 40 years. In the short run, FR-BDF’s dynamicscan deviate from long run targets due to nominal and real rigidities modeled with PAC. Asa result, convergence toward the BGP is achieved once both nominal and real rigidities havevanished. In particular, as explained in subsection 4.3 about the supply block, convergenceof the output gap toward 0 requires convergence of prices and employment toward theirtargets. To understand such a slow convergence, we should keep in mind two key featuresof the model, on which we will come back below: in the absence of independent monetarypolicy, the closure of these gaps is only ensured by price-competitiveness mechanisms whichaffect other model dynamics very slowly; the dynamics of these variables are also influencedby stock variables, i.e. capital services and net financial assets, which have very inertialdynamics.

More precisely, the path of the output gap goes through the following phases. At first, itincreases with a peak at almost 2% in 2020 mainly because of imports, which were initiallyabove their target and then return to their equilibrium level. On the nominal side, in additionto the upward pressure created by this increase in demand, as interest rates – initially atlow levels close to zero – converge toward values consistent with the steady state of theshort-run rate which is calibrated at its pre-crisis average (annualized rate at 3.7%), theprice target increases significantly with the cost of capital and the implied inflation resultsin a real appreciation (see Figure 5.1.2). This loss of price competitiveness pushes after 2020the output gap downward to such an extent that an undershoot occurs before its long run

73This version of the model is close to the one used for conditional forecasting, except for some differencesdetailed in section 4.11.

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Figure 5.1.1: Output and inflation gaps

-5

-4

-3

-2

-1

0

1

2

3

-5

-4

-3

-2

-1

0

1

2

3

2010 2020 2030 2040 2050 2060 2070 2080 2090 2100

Output gap Inflation gap

closure. Movements in the price level and inflation also amplify the effects of real interestrates on consumption and investment, because nominal interest rates are fixed, and slowdown convergence toward the BGP. As shown by Nakamura & Steinsson (2014), this channelis crucial in explaining the effects of demand shocks in monetary unions, in particular aftergovernment spending shocks.

Figure 5.1.2: Real effective exchange rate, PQ,t/Pcx,t

0.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04

1.06

0.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04

1.06

2010 2020 2030 2040 2050 2060 2070 2080 2090 2100

Finally, the convergence of net financial assets is illustrated in Figure 5.1.3. First, thanks

103

to stabilization rules described in section 4.8.5 about net asset positions, net asset ratiosof firms, non-profit institutions and government converge toward their steady state levels(respectively -70%, 2% and -40%, in percent of annualized GDP), although convergence isvery slow in the case of government net assets. It should be kept in mind that these stabi-lization rules are based on the gaps between agents’ net lending ratios and the correspondingdebt-stabilizing ratios. With such a specification and the chosen calibration, these rules arenon-aggressive, with the advantage of avoiding strong changes in the instruments in the shortrun and the drawback of such a slow convergence toward the long run.74 Second, as shown inthe same figure, households’ net asset ratio (and consequently of net foreign debt)convergestoward its steady state (around 120% in this simulation): this convergence is endogenouslyachieved in FR-BDF and critically depends on the marginal propensity to consume c.75 Ina simplified model of a small open economy, we can show that, in order to ensure thathouseholds do not accumulate wealth in a divergent way, households’ marginal propensity toconsume c should be larger than the threshold (i− g)/i, which depends on the steady stateof the nominal interest rate i and on the long run growth rate of nominal output g.76

Figure 5.1.3: Net financial assets, deviation from the steady-state level in pp of annualizedGDP

-60

-40

-20

0

20

40

60

-60

-40

-20

0

20

40

60

2050 2100 2150 2200 2250 2300

Non-financial and financial firms S11S12

General government S13

Households S14

Non-profit institutions serving households S15

74The empirical analysis of the types of instrument used in the past by the French government for debtstabilization and the aggressiveness of such policies is left for further research.

75Because of hysteresis implied by the usage of chained prices, this long-run ratio of net assets of householdsmight be sensitive to initial conditions of this simulation.

76A note with the detailed derivation of this condition in a simplified model is available upon request.

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5.2 Impulse responses

We turn to short run dynamic properties of FR-BDF and present four demand shocks andthree supply shocks. The four demand shocks are a standard monetary policy shock, aterm premium shock, a foreign demand shock and a public consumption shock. The firsttwo shocks will be analyzed briefly here and in more detail with a comparison of differentexpectations in section 6 as they are key to monetary policy transmission. The three supplyshocks are a cost-push shock on VA price, an oil price shock and a (permanent) laborefficiency shock. All simulations are performed around the balanced-growth path of themodel. We run an unconditional simulation from 2018Q1 to 2300Q1 which is our baseline,then we re-run an alternative simulation from 2149Q4 to 2300Q1 with the shock hittingthe economy in 2150Q1. IRFs are then calculated as percentages or absolute deviations,depending on the type of variable, between the alternative and baseline scenarios.77

5.2.1 Short-term interest rate shock

The monetary policy shock is designed as a one period +100bp shock to the annualizedshort-term interest rate which is endogenously passed on to the long-term rate through theterm structure equation and to the nominal effective exchange rate through the UIP. Theshock transmits to the model through it, the short-rate long-term expectation in E-SAT withan estimated historical persistence λi = 0.92; see section 3.1.2.

Figure 5.2.1 presents the short- to medium-term dynamics of core variables of FR-BDFin response to the short rate shock. First, as regards financial variables, we see that boththe long rate and the nominal effective exchange rate78 react one period after the shock, dueto backward-looking expectations: the long rate increases by +0.16pp, while the nominaleffective exchange rate appreciates by +0.40%. We observe a negative peak impact of -0.15%on real GDP, + 0.1pp on the unemployment rate and -0.10pp on the year-on-year VA priceinflation after 12 quarters. This shock also transmit directly through expectations, sincethe short rate is a core variable in E-SAT. On the real side, the short rate strongly affectsbusiness and household investments through the real user cost of capital, through its effecton expected inflation and on long-rates. It also affects the target for household consumptionboth through the real interest rate and through households’ expected income. Real exportsare also hit in the short run by the exchange rate appreciation. On the nominal side, the shortrate strongly affects nominal wage inflation and VA price respectively through the expectedunemployment gap and the expected change in VA price target inflation. Disinflation rapidlyimproves price competitiveness and the real effective exchange rate depreciates with respectto the baseline scenario after 12 quarters. Net exports significantly increase, boosting realGDP above the baseline after 24 quarters.

FR-BDF shows a stronger response than our former model Mascotte, which can be relatedto the stronger sensitivity of investment to the cost of capital and to the widespread influenceof the short-term rate through expectations, which play in particular a role here in term

77IRFs of inflation and interest rates as well as ratios with respect to GDP are calculated as absolutedeviations.

78In our IRFs, we report the nominal effective exchange rate of French exporters using indirect quotationrather than direct quotation (i.e. an increase of the nominal exchange rate means that the domestic currencyis appreciating).

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Figure 5.2.1: Short-term interest rate shock

-0.15

-0.10

-0.05

0.00

0.05

0.10

-0.15

-0.10

-0.05

0.00

0.05

0.10

4 8 12 16 20 24 28 32 36 40

Real GDP (% deviation)

-0.16

-0.12

-0.08

-0.04

0.00

0.04

-0.16

-0.12

-0.08

-0.04

0.00

0.04

4 8 12 16 20 24 28 32 36 40

Real private consumption (% deviation)

-0.80

-0.60

-0.40

-0.20

0.00

0.20

-0.80

-0.60

-0.40

-0.20

0.00

0.20

4 8 12 16 20 24 28 32 36 40

Real business investment (% deviation)

-0.80

-0.60

-0.40

-0.20

0.00

0.20

-0.80

-0.60

-0.40

-0.20

0.00

0.20

4 8 12 16 20 24 28 32 36 40

Real household investment (% deviation)

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

4 8 12 16 20 24 28 32 36 40

Real exports (% deviation)

-0.30

-0.20

-0.10

0.00

0.10

-0.30

-0.20

-0.10

0.00

0.10

4 8 12 16 20 24 28 32 36 40

Real imports (% deviation)

-0.02

0.00

0.02

0.04

0.06

0.08

-0.02

0.00

0.02

0.04

0.06

0.08

4 8 12 16 20 24 28 32 36 40

Trade balance/GDP (pp deviation)

-0.08

-0.04

0.00

0.04

0.08

0.12

-0.08

-0.04

0.00

0.04

0.08

0.12

4 8 12 16 20 24 28 32 36 40

Unemployment rate (pp deviation)

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

4 8 12 16 20 24 28 32 36 40

Consumption price inflation (y-o-y, pp deviation)

-0.12

-0.08

-0.04

0.00

0.04

-0.12

-0.08

-0.04

0.00

0.04

4 8 12 16 20 24 28 32 36 40

VA price inflation (y-o-y, pp deviation)

-0.16

-0.12

-0.08

-0.04

0.00

0.04

-0.16

-0.12

-0.08

-0.04

0.00

0.04

4 8 12 16 20 24 28 32 36 40

Wage inflation (y-o-y, pp deviation)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

4 8 12 16 20 24 28 32 36 40

Short-term nominal rate (y-o-y, pp deviation)

0.00

0.04

0.08

0.12

0.16

0.00

0.04

0.08

0.12

0.16

4 8 12 16 20 24 28 32 36 40

Long-term nominal rate (y-o-y, pp deviation)

0.00

0.10

0.20

0.30

0.40

0.50

0.00

0.10

0.20

0.30

0.40

0.50

4 8 12 16 20 24 28 32 36 40

Nominal effective exchange rate (% deviation)

-0.40

-0.20

0.00

0.20

0.40

0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

4 8 12 16 20 24 28 32 36 40

Real effective echange rate (% deviation)

structure and uncovered interest rate parity equations. Even with these features, the responseis weaker than in FRB/US: in addition to the absence of a reaction of the rest of the euroarea, which would have amplified the French response through foreign demand had it beenendogenous, this lower sensitivity could also be related to the absence of a wealth effect and afinancial accelerator in France, two particularly channels important for the United States.79

79Regarding the financial accelerator, this channel appears in FRB/US through an effect of the expectedoutput gap on risk premia. As we do not find a significant effect of this channel on French data, we do notinclude this channel in FR-BDF. Still, this result does not preclude that such a channel could play a role inFrance in stressed times. This topic is left for further research.

106

5.2.2 Term premium shock

Figure 5.2.2: Term-premium shock

-0.06

-0.04

-0.02

0.00

0.02

0.04

-0.06

-0.04

-0.02

0.00

0.02

0.04

4 8 12 16 20 24 28 32 36 40

Real GDP (% deviation)

-0.15

-0.10

-0.05

0.00

0.05

0.10

-0.15

-0.10

-0.05

0.00

0.05

0.10

4 8 12 16 20 24 28 32 36 40

Real private consumption (% deviation)

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

4 8 12 16 20 24 28 32 36 40

Real business investment (% deviation)

-0.30

-0.20

-0.10

0.00

0.10

-0.30

-0.20

-0.10

0.00

0.10

4 8 12 16 20 24 28 32 36 40

Real household investment (% deviation)

-0.04

-0.02

0.00

0.02

0.04

0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

4 8 12 16 20 24 28 32 36 40

Real exports (% deviation)

-0.12

-0.08

-0.04

0.00

0.04

0.08

-0.12

-0.08

-0.04

0.00

0.04

0.08

4 8 12 16 20 24 28 32 36 40

Real imports (% deviation)

-0.02

-0.01

0.00

0.01

0.02

0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

4 8 12 16 20 24 28 32 36 40

Trade balance/GDP (pp deviation)

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

4 8 12 16 20 24 28 32 36 40

Unemployment rate (pp deviation)

-0.03

-0.02

-0.01

0.00

0.01

0.02

-0.03

-0.02

-0.01

0.00

0.01

0.02

4 8 12 16 20 24 28 32 36 40

Consumption price inflation (y-o-y, pp deviation)

-0.03

-0.02

-0.01

0.00

0.01

0.02

-0.03

-0.02

-0.01

0.00

0.01

0.02

4 8 12 16 20 24 28 32 36 40

VA price inflation (y-o-y, pp deviation)

-0.03

-0.02

-0.01

0.00

0.01

0.02

-0.03

-0.02

-0.01

0.00

0.01

0.02

4 8 12 16 20 24 28 32 36 40

Wage inflation (y-o-y, pp deviation)

-1.00

0.00

1.00

-1.00

0.00

1.00

4 8 12 16 20 24 28 32 36 40

Short-term nominal rate (y-o-y, pp deviation)

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.20

0.40

0.60

0.80

1.00

4 8 12 16 20 24 28 32 36 40

Long-term nominal rate (y-o-y, pp deviation)

-1.00

0.00

1.00

-1.00

0.00

1.00

4 8 12 16 20 24 28 32 36 40

Nominal effective exchange rate (% deviation)

-0.06

-0.04

-0.02

0.00

0.02

0.04

-0.06

-0.04

-0.02

0.00

0.02

0.04

4 8 12 16 20 24 28 32 36 40

Real effective echange rate (% deviation)

The term premium shock is designed as a +100bp shock to the annualized nominal long-rate in the absence of any shock to the short-rate, with estimated historical persistence; seeTable 4.8.2 in section 4.8. Contrary to with the short rate shock, the long rate has no directeffect on expectations; in particular, the nominal effective exchange rate is unaffected bythe shock. The shock is endogenously passed on to bank-lending rates and to user costs ofcapital.

As shown in Figure 5.2.2, a +100bp term premium shock has a peak impact of -0.05%

107

on real GDP, +0.02pp on the unemployment rate and -0.025pp on VA price inflation. Onthe real side, the increase in the real user cost of capital for businesses and householdscauses a strong decrease in both types of investment, by -0.45% and -0.3% respectively. Onthe nominal side, Phillips effects cause a slowdown in VA price, wage and consumer priceinflation. The negative effect on inflation creates a real depreciation which in turn boostsnet exports in the short run. Contrary to the short rate shock, the term-premium shock hasa modest negative impact on real private consumption in the short run but a positive impactin the medium to long run, due to the increase in real income.

5.2.3 Foreign demand shock

The foreign demand shock is designed as a one-period +1% shock on the volume of foreigndemand addressed to French exporters, with a persistence calibrated at ρ = 0.9. We use thispersistence for shocks to exogenous variables.

Results are presented in Figure 5.2.3. After 4 quarters, the foreign demand shock has apeak impact of +0.14% on real GDP, -0.07pp on the unemployment rate and almost +0.075ppon VA price inflation. On the real side, the main short run driver of real GDP growth isnet exports, with the trade balance improving by +0.1pp of nominal GDP. Real exportsand imports increase respectively by +0.8% and +0.6% on impact. This strong response ofimports is related to two factors: (i) the large import content of exports (around 33%) takeninto account in the total import adjusted demand (IAD); (ii) the large short-run elasticityof imports excluding energy to IAD (1.9). Households’ private consumption and investmentalso increases on impact, by 0.03% and 0.04% respectively, due to the GDP growth gap intheir respective short run equations which account for rule-of-thumb agents. Finally, theshock has a peak impact of +0.2% on real business investment. On the nominal side, weobserve positive effects on both VA price and consumer price inflation respectively, due toPhillips effects in the short run VA price equation. After 10 quarters, wage inflation increasesby up to +0.06pp due to the decrease in the unemployment rate. Finally, the shock effecton real GDP turns negative after 24 quarters, due to the appreciation of the real effectiveexchange rate.

5.2.4 Government spending shock

The government spending shock is designed as an ex post +1% of GDP shock to impact onpublic consumption, with a persistence calibrated at ρ = 0.9.

As shown in Figure 5.2.4, real GDP increases by around +1.2% on impact and thenprogressively reverts to zero and turns negative after 18 quarters. Both unemployment andVA price inflation have hump-shaped responses to the shock. Unemployment decreases by-0.1pp on impact, with a peak impact of -0.45pp six quarters after the shock. VA price infla-tion increases by 0.1pp on impact and reaches +0.5pp after five quarters. On the real side,household consumption and investment increases on impact, by 0.3% and 0.4% respectively.Net exports decrease strongly on impact, through the direct effect on real imports but alsothrough the real exchange rate appreciation, due to higher VA price inflation. On the nom-inal side, the strong impact on VA price inflation and consumer price inflation reduces thereal user cost of capital for both households and businesses. More importantly, wages react

108

Figure 5.2.3: Foreign demand shock

-0.04

0.00

0.04

0.08

0.12

0.16

-0.04

0.00

0.04

0.08

0.12

0.16

4 8 12 16 20 24 28 32 36 40

Real GDP (% deviation)

0.00

0.02

0.04

0.06

0.08

0.10

0.00

0.02

0.04

0.06

0.08

0.10

4 8 12 16 20 24 28 32 36 40

Real private consumption (% deviation)

-0.05

0.00

0.05

0.10

0.15

0.20

-0.05

0.00

0.05

0.10

0.15

0.20

4 8 12 16 20 24 28 32 36 40

Real business investment (% deviation)

-0.10

-0.05

0.00

0.05

0.10

0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

4 8 12 16 20 24 28 32 36 40

Real household investment (% deviation)

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

4 8 12 16 20 24 28 32 36 40

Real exports (% deviation)

-0.20

0.00

0.20

0.40

0.60

-0.20

0.00

0.20

0.40

0.60

4 8 12 16 20 24 28 32 36 40

Real imports (% deviation)

-0.04

0.00

0.04

0.08

0.12

-0.04

0.00

0.04

0.08

0.12

4 8 12 16 20 24 28 32 36 40

Trade balance/GDP (pp deviation)

-0.08

-0.06

-0.04

-0.02

0.00

0.02

-0.08

-0.06

-0.04

-0.02

0.00

0.02

4 8 12 16 20 24 28 32 36 40

Unemployment rate (pp deviation)

-0.02

0.00

0.02

0.04

0.06

0.08

-0.02

0.00

0.02

0.04

0.06

0.08

4 8 12 16 20 24 28 32 36 40

Consumption price inflation (y-o-y, pp deviation)

-0.02

0.00

0.02

0.04

0.06

0.08

-0.02

0.00

0.02

0.04

0.06

0.08

4 8 12 16 20 24 28 32 36 40

VA price inflation (y-o-y, pp deviation)

-0.02

0.00

0.02

0.04

0.06

0.08

-0.02

0.00

0.02

0.04

0.06

0.08

4 8 12 16 20 24 28 32 36 40

Wage inflation (y-o-y, pp deviation)

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.20

0.40

0.60

0.80

1.00

4 8 12 16 20 24 28 32 36 40

Foreign demand (% deviation)

-1.00

0.00

1.00

-1.00

0.00

1.00

4 8 12 16 20 24 28 32 36 40

Short-term nominal rate (y-o-y, pp deviation)

-1.00

0.00

1.00

-1.00

0.00

1.00

4 8 12 16 20 24 28 32 36 40

Nominal effective exchange rate (% deviation)

0.00

0.05

0.10

0.15

0.20

0.25

0.00

0.05

0.10

0.15

0.20

0.25

4 8 12 16 20 24 28 32 36 40

Real effective echange rate (% deviation)

less than both VA price and consumer price inflation, which reduces the real labour costand boosts employment through demand in the short run, but also reduces real householdincome and consumption 16 quarters after the shock.

The size of the government spending multiplier, above 1 at impact in FR-BDF, is broadlyin line with those found in the empirical literature. Notable features are a positive crowding-in effect on both consumption and investment in the short run. Empirical literature usuallyfinds higher multipliers in monetary unions or fixed-exchange rate economies, because thenominal interest rate does not react (much) to regional or national inflation (Nakamura &Steinsson, 2014). More generally, fiscal multipliers can be much higher when monetary policy

109

Figure 5.2.4: Government spending shock

-0.40

0.00

0.40

0.80

1.20

-0.40

0.00

0.40

0.80

1.20

4 8 12 16 20 24 28 32 36 40

Real GDP (% deviation)

-0.40

-0.20

0.00

0.20

0.40

-0.40

-0.20

0.00

0.20

0.40

4 8 12 16 20 24 28 32 36 40

Real private consumption (% deviation)

-0.40

0.00

0.40

0.80

1.20

1.60

-0.40

0.00

0.40

0.80

1.20

1.60

4 8 12 16 20 24 28 32 36 40

Real business investment (% deviation)

-0.80

-0.40

0.00

0.40

0.80

1.20

-0.80

-0.40

0.00

0.40

0.80

1.20

4 8 12 16 20 24 28 32 36 40

Real household investment (% deviation)

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

4 8 12 16 20 24 28 32 36 40

Real exports (% deviation)

-0.40

0.00

0.40

0.80

1.20

-0.40

0.00

0.40

0.80

1.20

4 8 12 16 20 24 28 32 36 40

Real imports (% deviation)

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

4 8 12 16 20 24 28 32 36 40

Trade balance/GDP (pp deviation)

-0.60

-0.40

-0.20

0.00

0.20

-0.60

-0.40

-0.20

0.00

0.20

4 8 12 16 20 24 28 32 36 40

Unemployment rate (pp deviation)

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

4 8 12 16 20 24 28 32 36 40

Consumption price inflation (y-o-y, pp deviation)

-0.20

0.00

0.20

0.40

0.60

-0.20

0.00

0.20

0.40

0.60

4 8 12 16 20 24 28 32 36 40

VA price inflation (y-o-y, pp deviation)

-0.20

0.00

0.20

0.40

0.60

-0.20

0.00

0.20

0.40

0.60

4 8 12 16 20 24 28 32 36 40

Wage inflation (y-o-y, pp deviation)

-1.00

0.00

1.00

-1.00

0.00

1.00

4 8 12 16 20 24 28 32 36 40

Short-term nominal rate (y-o-y, pp deviation)

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.20

0.40

0.60

0.80

1.00

4 8 12 16 20 24 28 32 36 40

Real effective exchange rate (% deviation)

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.20

0.40

0.60

0.80

1.00

4 8 12 16 20 24 28 32 36 40

Government consumption/GDP (pp deviation)

-0.20

-0.16

-0.12

-0.08

-0.04

-0.20

-0.16

-0.12

-0.08

-0.04

4 8 12 16 20 24 28 32 36 40

Social transfers/long-run GDP (pp deviation)

is "passive" or at the zero lower bound (Leeper et al., 2017).In contrast, when Farhi & Werning (2016) assume that households are Ricardian and

government spending is financed by a budget-neutral rule on lump-sum transfers, they findsome crowding out in an open economy DSGE model in monetary union, because of thereal appreciation plus expected reversal of the price level. However, in an other case, theyassume the presence of non-Ricardian households and outside-financed government spending,which leads to crowding in. This case seems more similar to our setup, as we have: (i) non-Ricardian effects in the short-run equation of consumption thanks to the inclusion of thecurrent change of the output gap; (ii) non-aggressive fiscal rule on social transfers.

110

5.2.5 Oil price shock

Figure 5.2.5: Oil price shock

-0.20

-0.15

-0.10

-0.05

0.00

0.05

-0.20

-0.15

-0.10

-0.05

0.00

0.05

4 8 12 16 20 24 28 32 36 40

Real GDP (% deviation)

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

4 8 12 16 20 24 28 32 36 40

Real private consumption (% deviation)

-0.40

-0.30

-0.20

-0.10

0.00

0.10

-0.40

-0.30

-0.20

-0.10

0.00

0.10

4 8 12 16 20 24 28 32 36 40

Real business investment (% deviation)

-0.16

-0.12

-0.08

-0.04

0.00

-0.16

-0.12

-0.08

-0.04

0.00

4 8 12 16 20 24 28 32 36 40

Real household investment (% deviation)

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

4 8 12 16 20 24 28 32 36 40

Real exports (% deviation)

-0.16

-0.12

-0.08

-0.04

0.00

-0.16

-0.12

-0.08

-0.04

0.00

4 8 12 16 20 24 28 32 36 40

Real imports (% deviation)

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

4 8 12 16 20 24 28 32 36 40

Trade balance/GDP (pp deviation)

-0.04

0.00

0.04

0.08

0.12

-0.04

0.00

0.04

0.08

0.12

4 8 12 16 20 24 28 32 36 40

Unemployment rate (pp deviation)

-0.20

-0.10

0.00

0.10

0.20

0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

4 8 12 16 20 24 28 32 36 40

Consumption price inflation (y-o-y, pp deviation)

-0.08

-0.06

-0.04

-0.02

0.00

0.02

-0.08

-0.06

-0.04

-0.02

0.00

0.02

4 8 12 16 20 24 28 32 36 40

VA price inflation (y-o-y, pp deviation)

-0.15

-0.10

-0.05

0.00

0.05

0.10

-0.15

-0.10

-0.05

0.00

0.05

0.10

4 8 12 16 20 24 28 32 36 40

Wage inflation (y-o-y, pp deviation)

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0.00

2.00

4.00

6.00

8.00

10.00

12.00

4 8 12 16 20 24 28 32 36 40

Brent price (% deviation)

-1.00

0.00

1.00

-1.00

0.00

1.00

4 8 12 16 20 24 28 32 36 40

Short-term nominal rate (y-o-y, pp deviation)

-1.00

0.00

1.00

-1.00

0.00

1.00

4 8 12 16 20 24 28 32 36 40

Nominal effective exchange rate (% deviation)

-0.40

-0.30

-0.20

-0.10

0.00

-0.40

-0.30

-0.20

-0.10

0.00

4 8 12 16 20 24 28 32 36 40

Real effective echange rate (% deviation)

The oil price shock is designed as a +10% shock to the Brent oil price (in 2014 euros,the shock is equivalent to +4.6 euros), during 8 quarters, with a persistence calibrated atρ = 0.9 afterward. More importantly, the shock affects the economy through the energyimport price. There is no direct effect on the VA price through the Factor Price Frontier(FPF). Finally, foreign competitors’ prices are left unchanged, i.e. the shock is asymmetric.

Results are shown in Figure 5.2.5. After 12 quarters, the shock has a peak impact of -0.2%on real GDP, +0.1pp on unemployment and -0.06pp on VA price inflation. On the nominal

111

side, the main effect occurs through the energy import price and consumer inflation, with apeak impact of +0.25pp on the latter. At first, wages react incompletely and with a lag to theincrease in consumer price inflation, which will reduce real household income in the short run.Wage inflation then declines by -0.12pp after 16 quarters, due to the rise in unemployment.In contrast, the response of VA price inflation shows an absence of second-round effects: VAprice inflation does not react on impact but will decrease thereafter by 0.06pp, due to Phillipseffects from the real side. On the one hand, households’ purchasing power will decline in theshort run, due to the strong increase in the consumer price, while on the other, the real laborcost will rise and reduce labor-demand. On the real side, all aggregate demand componentsfall after the shock. Household consumption and investment decrease progressively due tolower purchasing power and higher unemployment. Real business investment decreases dueto the decline in the VA of market branches. Finally, net exports decline in the short run(-0.25% after 10 quarters) mostly due to the import content of exports.80 Meanwhile, realimports only fall progressively to -0.12%, along with aggregate demand components. In themedium run, the decrease in the VA price progressively restores price competitiveness andreal exports rise above the baseline after 22 quarters.

5.2.6 Cost-push shock

Our cost-push shock is designed as a markup shock such that the VA price increases by 1%on impact and is then endogenously passed on to the economy according to the persistenceof the VA price short run equation. As the VA price is a core variable in E-SAT, the shockwill affect PAC equations through expectation terms.81

Results are shown in Figure 5.2.6. On impact, the shock materializes by a +1pp increasein VA price inflation. After a peak impact of +1.5pp after four quarters, VA price inflationrapidly decreases and stays below the baseline during around 20 quarters. It has a peakimpact of -0.45% on real GDP after eight quarters and +0.2pp on unemployment after 12quarters. On the nominal side, consumer price inflation follows a similar path as VA priceinflation: it increases by +0.6pp on impact and rapidly decreases to 0 after 64 quarters.Lower inflation with respect to the baseline after four quarters is necessary for the pricelevel to adjust downward and for the output gap to close through external trade (see below).As after the oil price shock, wage inflation underreacts to consumer price inflation due toincomplete indexation in the wage Phillips-curve. On the real side, household consumptionand investment are strongly and negatively affected by the decrease in expected real income.Both sub-components of real investment display a similar response. At first, their responsesare dampened by the fall in the real user cost of capital, implied by higher inflation. Thenegative effect on market branches’ VA and on households real expected income then reducesboth types of investment demand.82 As regards the external trade sector, the trade balance

80The export price strongly depends on the import price both in the long run and in the short run, dueto the import content of exports. As a result, a shock to the energy import price increases the export priceand reduces the price competitiveness of exports.

81The shock is technically engineered as a one-period 1% shock on the residual of VA price short runequation.

82The stronger effect on household investment is due to the higher persistence in the household investmentequation compared to business investment, which amplifies the initial shock.

112

Figure 5.2.6: Cost-push shock

-0.50

-0.40

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0.00

0.10

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0.10

4 8 12 16 20 24 28 32 36 40

Real GDP (% deviation)

-0.30

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-0.10

0.00

0.10

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-0.10

0.00

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4 8 12 16 20 24 28 32 36 40

Real private consumption (% deviation)

-0.30

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-0.10

0.00

0.10

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-0.10

0.00

0.10

4 8 12 16 20 24 28 32 36 40

Real business investment (% deviation)

-0.80

-0.60

-0.40

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0.00

0.20

-0.80

-0.60

-0.40

-0.20

0.00

0.20

4 8 12 16 20 24 28 32 36 40

Real household investment (% deviation)

-1.20

-0.80

-0.40

0.00

0.40

-1.20

-0.80

-0.40

0.00

0.40

4 8 12 16 20 24 28 32 36 40

Real exports (% deviation)

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

4 8 12 16 20 24 28 32 36 40

Real imports (% deviation)

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

4 8 12 16 20 24 28 32 36 40

Trade balance/GDP (pp deviation)

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

4 8 12 16 20 24 28 32 36 40

Unemployment rate (pp deviation)

-1.00

-0.50

0.00

0.50

1.00

1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

4 8 12 16 20 24 28 32 36 40

Consumption price inflation (y-o-y, pp deviation)

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

4 8 12 16 20 24 28 32 36 40

VA price inflation (y-o-y, pp deviation)

-0.40

-0.20

0.00

0.20

0.40

0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

4 8 12 16 20 24 28 32 36 40

Wage inflation (y-o-y, pp deviation)

-1.00

0.00

1.00

-1.00

0.00

1.00

4 8 12 16 20 24 28 32 36 40

Short-term nominal rate (y-o-y, pp deviation)

-1.00

0.00

1.00

-1.00

0.00

1.00

4 8 12 16 20 24 28 32 36 40

Long-term nominal rate (y-o-y, pp deviation)

-1.00

0.00

1.00

-1.00

0.00

1.00

4 8 12 16 20 24 28 32 36 40

Nominal effective exchange rate (% deviation)

-0.40

0.00

0.40

0.80

1.20

1.60

-0.40

0.00

0.40

0.80

1.20

1.60

4 8 12 16 20 24 28 32 36 40

Real effective echange rate (% deviation)

temporarily increases after the shock, as a result of the real exchange rate appreciation effecton imports but then decreases after four quarters. Finally, the real effective exchange ratedepreciation progressively restores price competitiveness and net exports, which boosts realGDP.

113

5.2.7 Labor efficiency shock

The labor efficiency shock is designed as a permanent +1% shock to the level of trend laborefficiency, with estimated historical persistence.83 The shock’s main transmission channelsare long run output, VA price and labor demand targets. Through its impact on long runouput and on the output gap, it also affects all expectations terms. Given that it is apermanent shock to the level of trend labor efficiency, it naturally has permanent effectswith respect to the baseline scenario. Still, the growth rate of long run output is unchangedat the BGP. As a result, IRFs describe the transitory dynamics toward the new BGP.

Figure 5.2.7 shows the results. Initially, the shock has a positive and progressive impacton real GDP and long run real GDP, with a larger effect on the latter resulting in a neg-ative output gap. The productivity shock reduces labor demand, leading to an increase inunemployment. It then progressively declines with the rise in output. On the real side, allcomponents of domestic demand progressively increase. Long run real GDP directly drivesexpected real income and both household consumption and investment. Business investmentreacts with lags to the shock, mostly due to a higher real user cost of capital, but progres-sively rises with market branches’ real value-added. The trade balance first deteriorates onaccount of the increase in imports, in line with domestic demand. But real exports thenprogressively increase thanks to the real exchange rate depreciation. On the nominal side,the productivity shock reduces VA price inflation by -0.09pp and consumer price inflationby -0.07pp after eight quarters. In the meantime, wage inflation rises by about +0.2pp onimpact (up to +0.6pp four quarters after the shock) because wages increase one-to-one withtrend labor efficiency in our Wage-PC (see equation 52).

Finally, the real exchange rate continuously depreciates: following the shock, the domesticprice level is permanently reduced. In particular, VA price inflation stays persistently belowthe baseline up to 40 quarters, due to high inertia of employment. As a result, the positiveunemployment gap slows wage inflation persistently by around -0.05pp 20 quarters after theshock, which explains why inflation remains below the baseline although the output gap hasclosed.

83The shock is implemented as a permanent shock to the residual of the trend labor efficiency equation,described in section 4.3.

114

Figure 5.2.7: Labor efficiency shock

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.20

0.40

0.60

0.80

1.00

4 8 12 16 20 24 28 32 36 40

Real GDP (% deviation)

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.20

0.40

0.60

0.80

1.00

4 8 12 16 20 24 28 32 36 40

Long-run real GDP (% deviation)

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

4 8 12 16 20 24 28 32 36 40

Output gap (pp deviation)

0.00

0.20

0.40

0.60

0.80

0.00

0.20

0.40

0.60

0.80

4 8 12 16 20 24 28 32 36 40

Real private consumption (% deviation)

0.00

0.20

0.40

0.60

0.80

0.00

0.20

0.40

0.60

0.80

4 8 12 16 20 24 28 32 36 40

Real business investment (% deviation)

-0.20

0.00

0.20

0.40

0.60

0.80

-0.20

0.00

0.20

0.40

0.60

0.80

4 8 12 16 20 24 28 32 36 40

Real household investment (% deviation)

0.00

0.10

0.20

0.30

0.40

0.00

0.10

0.20

0.30

0.40

4 8 12 16 20 24 28 32 36 40

Real exports (% deviation)

0.10

0.13

0.15

0.18

0.20

0.22

0.25

0.10

0.13

0.15

0.18

0.20

0.22

0.25

4 8 12 16 20 24 28 32 36 40

Real imports (% deviation)

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

4 8 12 16 20 24 28 32 36 40

Trade balance/GDP (pp deviation)

0.00

0.04

0.08

0.12

0.16

0.00

0.04

0.08

0.12

0.16

4 8 12 16 20 24 28 32 36 40

Unemployment rate (pp deviation)

-0.08

-0.06

-0.04

-0.02

0.00

-0.08

-0.06

-0.04

-0.02

0.00

4 8 12 16 20 24 28 32 36 40

Consumption price inflation (y-o-y, pp deviation)

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

4 8 12 16 20 24 28 32 36 40

VA price inflation (y-o-y, pp deviation)

-0.20

0.00

0.20

0.40

0.60

0.80

-0.20

0.00

0.20

0.40

0.60

0.80

4 8 12 16 20 24 28 32 36 40

Wage inflation (y-o-y, pp deviation)

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

4 8 12 16 20 24 28 32 36 40

Real effective exchange rate (% deviation)

0.20

0.40

0.60

0.80

1.00

0.20

0.40

0.60

0.80

1.00

4 8 12 16 20 24 28 32 36 40

Trend labor efficiency (% deviation)

115

6 Monetary policy transmission under different expecta-tion assumptions

6.1 Alternative version of FR-BDF with model-consistent expecta-tions

We apply FR-BDF with model-consistent expectations for two purposes: (i) to better un-derstand the VAR-based case and the behavior of the model; and (ii) to study policy issueswhere forward-looking behavior matters, as in e.g. a forward guidance exercise.84

6.1.1 Equations of present value variables under model-consistent expectations

First, we present the equations for the present values with a constant discounting scheme.85

They can be written in the forward-looking way with a single lead, see subsection 3.3.2. Thediscounting weights sum to unity in all cases, except for the present value of the expectednon-discounted sums of future short run interest rates and future foreign short run interestrates.

PV (i)t = (1− 0.97)it + 0.97PV (i)t+1 (132)PVnd(i− i)t = it − i+ PVnd(i− i)t+1 (133)PVnd(iF − i)t = iF,t − i+ PVnd(iF − i)t+1 (134)PV (πQ)t = (1− 0.994)πQ,t + 0.994PV (πQ)t+1 (135)

PV (yH)t = (1− 0.95)yH,t +0.95

exp(∆y)PV (yH)t+1 (136)

PV (u)t = (1− 0.98)ut + 0.98PV (u)t+1 (137)

Second, we present equations for the present values of expected changes in targets whichappear in PAC equations. Their parameters are determined by the structure and parametersof the corresponding short run equation and discount factor, see subsection 3.3.1.86 The

84For forecasting only VAR expectations are applied as it is easier to apply judgment.85The choice of discount factors is explained in the sections of the corresponding variables, i.e. in the

subsection 4.8 about the financial block for expected interest rates, in the subsection 4.6.2 about businessinvestment for expected inflation, in the subsection 4.6.1 about consumption for expected income and in thesubsection 4.5.1 about labor supply for expected unemployment.

86The discount factor of PAC equations is generally calibrated at 0.98, as explained in the subsection 4.1about the estimation approach.

116

expectation is defined as a weighted sum of a variable over future quarters.

PV (π∗Q)t = β0 π∗Q,t + β1π∗Q,t+1 + β2PV (π∗Q)t+1 + β3PV (π∗Q)t+2 (138)

PV (∆n∗S)t = β0 ∆n∗S,t + β1∆n∗S,t+1 + β2∆n∗S,t+2 + β3∆n∗S,t+3 + β4PV (∆n∗S)t+1

+ β5PV (∆n∗S)t+2 + β6PV (∆n∗S)t+3 + β7PV (∆n∗S)t+4 (139)PV (∆c∗)t = β0 ∆c∗t + β1∆c∗t+1 + β2PV (∆c∗)t+1 + β3PV (∆c∗)t+2 (140)

PV (∆logI∗H)t = β0 ∆logI∗H,t + β1∆logI∗H,t+1 + β2PV (∆logI∗H)t+1

+ β3PV (∆logI∗H)t+2 (141)PV (∆logI∗B)t = β0 ∆logI∗B,t + β1∆logI∗B,t+1 + β2∆logI∗B,t+2 (142)

+ β3PV (∆logI∗B)t+1 + β4PV (∆logI∗B)t+2 + β5PV (∆logI∗B)t+3

The estimated parameters are shown in Table 6.1.1.

Table 6.1.1: Coefficients of equations of present values in the model-consistent case

PV (π∗Q)t PV (∆n∗S)t PV (∆c∗)t PV (∆logI∗H)t PV (∆logI∗B)tCoef. Coef. Coef. Coef. Coef.

β0 0.06 0.06 0.12 0.06 0.08β1 -0.03 -0.05 0.01 -0.03 -0.02β2 1.41 0.02 0.75 1.53 -0.02β3 -0.48 -0.01 0.07 -0.59 1.18β4 - 1.77 - - -0.09β5 - -1.11 - - -0.18β6 - 0.45 - - -β7 - -0.17 - - -

Implementation In order to simulate the FR-BDF model under MCE, we first compilethe VAR-based version, then remove the objects that stand for expectations and replacethem with the equations above. For example, in the case of the consumption targer, in orderto move from VAR-based case to MCE case, we remove the following expression

PV2 (yH − y)t|t−1 + α1

(PV (iH)t|t−1 −

(PV (i)t|t−1 − PV (π)t|t−1

))from the short run equation of household consumption (62) and replace it with equation(140).

6.1.2 Simulation methodology

Counterfactual experiments under model-consistent expectations are computed in deviationfrom a baseline because FR-BDF is a non-linear model.87 In order to be able to compare cases

87An example of non-linearity is the application of the log operator to the cost of capital.

117

under different expectation assumptions the baselines should be identical. This conditiondefines that the procedure we apply for unconditional simulations in the MCE case: it isdone by inverting the MCE model and solving for "new" expectations and residuals on thehistorical sample, leaving observed variables unchanged. It should also be mentioned herethat the MCE version is solved under perfect foresight and we do not take into account anyuncertainty.

6.2 Monetary policy shock

Figure 6.2.1: Annualized short run interest rate after the monetary policy shock

-1.2

-1.0

-0.8

-0.6

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0.0

-1.2

-1.0

-0.8

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-0.2

0.0

1 21 41 61 81

MCE and Hyb.E VAR-based

Short-run interest rate (annualized)

The exercise In this section, we consider the response of FR-BDF to a 100bp shock tothe annualized short run interest rate. The shock occurs in the future, sufficiently far aheadin order to consider that the model is at the steady state. The shock is temporary and lastsone quarter, see Figure 6.2.1. The short run interest rate follows an AR(1) process witha persistence parameter of 0.92, the one of the Taylor rule estimated with the expectationsatellite model E-SAT. The euro area variables are exogenous.88 In this section all theresponses are presented in deviation from a baseline in pp.

This exercise is simulated under three expectation assumptions: (i) agents constructtheir expectations using the E-SAT model (VAR-based case), (ii) agents are forward-looking(MCE case) and (iii) Hybrid case (Hyb.E), where financial agents are forward-looking, i.e.,their expectations are model-consistent, and the others use limited information to predictthe future, i.e., their expectations are VAR-based. To be more specific, financial variablesthat are forward-looking in the Hybrid case include (i) the expected sum of discounted shortrun interest rates (which influences the long run interest rate), (ii) the expected sum of non-discounted short run interest rates (which influences the exchange rate), (iii) the expectedsum of non-discounted foreign short run interest rates (which influences the exchange rate).89

Table 6.2.1 summarizes the differences between the expectation assumptions.88This is a strong yet important assumption, as euro area variables enter E-SAT and are hence included

in the policy function of expected variables in the VAR-based case.89However, the latter is exogenous and does not matter for the exercise.

118

Table 6.2.1: Differences between three expectation assumptions

Financial variables Non-financial variablesHybrid forward-looking backward-lookingMCE forward-looking forward-looking

VAR-based backward-looking backward-looking

The propagation mechanism is the same under all expectation assumptions. First, adecline in the short run interest rate results in a decrease in the long run interest rate,which is the main driver of the bank lending rate and cost of capital, encouraging agentsto invest. As regards external trade, exports rise due to the depreciation in the domesticcurrency caused by the difference between domestic and foreign short run interest rates.This, in turn, leads to a boom in the labor market and an increase in real disposable income– one of the main drivers of household consumption. The price response is positive and verypersistent, which leads to a loss in competitiveness, decreasing exports in the medium runand downward pressure on the economy.

Figure 6.2.2: Monetary policy responses under different types of expectations

-.2

-.1

.0

.1

.2

.3

-.2

-.1

.0

.1

.2

.3

1 11 21 31 41 51 61 71 81

VAR-based Hyb.E. MCE

Quarters

Output

(deviation from baseline, in %)

-.10

-.05

.00

.05

.10

.15

.20

-.10

-.05

.00

.05

.10

.15

.20

1 11 21 31 41 51 61 71 81

VAR-based Hyb.E. MCE

Quarters

VA price inflation

(annualized, deviation from baseline, in pp)

Main results The output and the value-added inflation responses are shown in Figure6.2.2, the GDP components are available in Figure A.2 in the Appendix. All three cases aredifferent in amplitude and speed of convergence. Three conclusions can be reached at thisstage. First, we observe that the GDP impact response in the full MCE case is below thatin the Hybrid case. This leads us to conclude that when non-financial variables are modeledin a forward-looking manner, they create a dampening effect in comparison to a VAR-basedcase. Second, the GDP response in the Hybrid case is larger than in the VAR-based case.This leads us to believe that forward-looking financial variables create an amplification effect

119

in comparison to the VAR-based case. Third, convergence is much slower in the cases wherenon-financial variables are backward-looking with a long-lasting "undershoot" of demandcomponents (VAR-based and Hybrid cases). The rest of this section discusses these pointsin detail.

This last difference in responses to an interest rate shock (long-lasting undershoot underVAR-based expectations but not under MCE) is similar to a difference pointed out in Roeger& Herz (2012) in the case of a closed-economy model with a Phillips curve that is eitherforward-looking or backward-looking and accelerationist.90

Dampening effect of forward-looking non-financial variables. The reason for thisdampening effect is that backward-looking agents are more optimistic. They form theirexpectations using E-SAT, which predicts higher economic growth after the same shock. InE-SAT, the output gap reaches 0.34% at the peak (see Figure 3.1.2), while in the MCE case,it stands at 0.17% at the peak. In order to understand the implication of this "optimism",we compare responses under the Hybrid and MCE cases (see Table 6.2.1 for a reminder onthe differences between them).

Figure 6.2.3: Monetary policy responses – 1st focus on Hyb.E. versus MCE

-.08

-.06

-.04

-.02

.00

.02

.04

-.08

-.06

-.04

-.02

.00

.02

.04

1 21 41 61 81 101 121

Hyb.E. MCE

Quarters

Expected sum of future unemployment gaps

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1 21 41 61 81 101 121

Hyb.E. MCE

Quarters

Wage per head (market branches)

-.1

.0

.1

.2

.3

.4

.5

.6

-.1

.0

.1

.2

.3

.4

.5

.6

1 21 41 61 81 101 121

Hyb.E. MCE

Quarters

Consumption price

-.6

-.4

-.2

.0

.2

.4

.6

.8

-.6

-.4

-.2

.0

.2

.4

.6

.8

1 21 41 61 81 101 121

Hyb.E. MCE

Quarters

Exports

-.16

-.12

-.08

-.04

.00

.04

.08

.12

.16

.20

-.16

-.12

-.08

-.04

.00

.04

.08

.12

.16

.20

1 21 41 61 81 101 121

Hyb.E. MCE

Quarters

Expected real disposable income (volume)

-.2

-.1

.0

.1

.2

.3

-.2

-.1

.0

.1

.2

.3

1 21 41 61 81 101 121

Hyb.E. MCE

Quarters

HH consumption target

90In this paper, authors note that the forward-looking case would be more realistic for the United States,given that they do net find an overshoot in impulse responses of a SVAR estimated on US data. Our setupis quite different: we have an anchored Phillips curve and an open economy setup in a monetary union. Itis unclear for us at this stage which response would be empirically more appropriate on French data.

120

The higher growth expected by the VAR-based non-financial agents (Hybrid case) leads to(i) lower expectations regarding the unemployment gap and (ii) higher expectations regardingthe value-added inflation in comparison to the MCE case. Acting through the wage Phillipscurve, lower expectations regarding the unemployment gap imply higher wages per capitaand – via the price/wage loop – higher price level, see Figure 6.2.3.91 In the short run, thisleads to a larger increase in expected disposable income in the Hybrid case in comparisonto the MCE case, which explains the differences in household consumption targets since thelatter is one of their main components, see Figure A.2.92 In the medium run since the pricelevel stays positive, there is a loss in competitiveness and, hence, a decrease in exports. Inthe Hybrid case, the loss is so significant that it produces undershooting in exports and,through the labor market, in all the demand components. In the MCE case forward-lookingagents are aware of the future loss in income flows. They start smoothing their consumptionfrom the beginning (see the comparison of the expected disposable income in Figure 6.2.3).

Figure 6.2.4: Monetary policy responses – 2nd focus on Hyb.E. versus MCE

-.01

.00

.01

.02

.03

.04

.05

-.01

.00

.01

.02

.03

.04

.05

1 21 41 61 81 101 121

Hyb.E. MCE

Quarters

Expected VA inflation

-.10

-.08

-.06

-.04

-.02

.00

.02

-.10

-.08

-.06

-.04

-.02

.00

.02

1 21 41 61 81

Hyb.E. MCE

Quarters

Real bank lending rate for HH

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1 21 41 61 81

Hyb.E. MCE

Quarters

Real cost of capital (SNF SF EI )

The larger positive response of expected value added inflation in the Hybrid case explainsthe difference in business and household investment when non-financial agents have differentexpectation assumptions, see Figure 6.2.4. Higher expected value-added inflation pushes thereal bank lending rate and the real cost of capital down, while their nominal counterpartsdisplay the same response in both cases since the long run interest rate response is the samedue to the fact that the financial variables share the same expectation formation.

Amplification effect of forward-looking financial variables. Model-consistent expec-tations of the short run interest rate are more reactive than backward-looking expectations

91Similar results were found in Mankiw & Reis (2002). Backward-looking price setting delivers a moreinertial pattern of inflation compared to forward-looking behavior. Mankiw & Reis (2002), in particular,emphasize this difference between the New Keynesian Phillips Curve (which associates sticky prices andrational expectations) and the "backward-looking" Phillips Curve (associating sticky prices and adaptiveexpectations.

92Note that in the short run employment behaves similarly in both cases.

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mainly because of the assumption that euro area variables are exogenous.93 Indeed, underVAR-based expectations, even if EA variables are exogenized, the sensitivity of expectationcomponents to the long run interest rate and exchange rates to the euro area short run ratecomes from our VAR model (E-SAT), where the euro area is endogenous. In this VAR, acut of the short run interest rate generates a boom of the output gap and inflation and,hence, a monetary policy tightening in the medium run. As agents expect this tightening,expectation components, which are loosely speaking averages of future short run rates, de-crease less than if the short run rate would have reverted with historical persistence withoutany endogenous response of monetary policy to macroeconomic conditions (which is whathappens under model-consistent expectations).

The difference in the short run interest rate expectations leads to differences in expec-tations regarding (i) the long run interest rate and (ii) the exchange rate, see Figure 6.2.5.Since the long run interest rate is the main driver of the nominal bank lending rate and theWACC, investment is more attractive in the case of forward-looking expectations of financialvariables (Hybrid case).94 Stronger depreciation in the Hybrid case boosts exports and anincrease in the latter also contributes to the boom in the labor market and income flows.Note here that the difference in real exchange rates is mainly explained by the differencein the nominal rate. However, due to a higher cost of capital in the Hybrid case, the priceincrease is also stronger, which further amplifies the difference in real exchange rates.

6.3 Forward guidance95

6.3.1 Theoretical background for the forward guidance puzzle

Baseline DSGE models generically predict a very strong reaction of output and inflation toforward guidance policies. In addition, they predict that the further away the date when theinterest rate is promised to decrease, the larger the immediate increase in GDP and inflation.If anything, one would expect promises of future interest rate cuts to be less powerful thancurrent interest-rate cuts. This peculiar prediction of DSGE models has come to be knownas the forward guidance puzzle, as coined by Del Negro et al. (2012). The puzzle statesthat the further away the date when the interest rate is promised to increase, the larger theimmediate effect on household behavior and thus on inflation. In this section, we investigatethis phenomenon using FR-BDF and a pair of DSGE models:

• NK 3-equation model both under a standard calibration (denoted "nondiscountedNK") and an alternative one (denoted "discounted NK") which assumes that house-

93Even though we have equations for the euro area variables, the foreign block does not react to themand we do not wish to simulate the model with endogenous euro area variables in a part of it (expectationvariables) while keeping them exogenous in other parts (foreign demand and prices). In order to fullyendogenize the role of the euro area, we would need to connect the model to a model of the euro area.In this case the model would retain the same expectations irrespective of whether financial variables areforward-looking or not. This exercise is left for further research. Meanwhile, we analyze the effect of theassumption of exogenous euro area variables on our results and, to do so, we compare the VAR-based casewith the Hybrid case.

94 Note that inflation expectations are more or less the same in both cases (they are backward-looking).95We would like to thank Julien Matheron and Stephane Dupraz for providing the simulation results for

the FPH model and for offering helpful comments.

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Figure 6.2.5: Monetary policy responses – focus on Hyb.E. versus VAR-based

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holds do not react as much to interest-rate cuts far in the future because they discountfuture consumption, which McKay et al. (2017) argue could arise e.g. if some house-holds are credit constrained. This model is characterized by equations (143), (144) andan inertial Taylor rule.

• The DSGE model of Woodford & Xie (2019) where agents have finite planning horizons,denoted FPH. The average planning horizon is set to four quarters.

McKay et al. (2016) provide a clear explanation based on the intertemporal IS and Phillipscurves. For the sake of simplicity we provide expressions for these curves below. A linearizedform of the Euler equation, as presented in e.g. Galí (2008) is given by:

ct = Etct+1 − σ1 (rt − r) (143)

where rt is the real short rate and ct stands for consumption. Note that it is common torewrite (143) with respect to the output gap xt instead of ct to obtain what is called theintertemporal IS curve. In a more complex model, such as the one in Smets & Wouters(2007), the equation would contain additional terms such as lagged consumption due tohabit formation in utility and a term measuring the disutility of labor. For the simulationexercise we modify this equation in the alternative case ("discounted NK") by multiplyingthe first term on the right hand side by a discount factor ν = 0.97.

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The New Keynesian Phillips curve relates inflation to the output gap and expected in-flation. The standard formulation is:

πt = β1Etπt+1 + κxt (144)

As with Euler equations, more complicated models have more complicated analogousexpressions, e.g. in Smets & Wouters (2007) an additional lagged inflation term appears inthe equation due to indexation of prices. To account for rigidities on the labor market, theyalso have a wage Phillips curve.

First, assuming that the real interest rate is constant, note that the Phillips curve (144)can be re-expressed as a discounted infinite sum of future output gaps and the IS curve (143)as the sum of future short rates:

πt = κ∞∑j=0

βjEtxt+j,

xt = −σ1

∞∑j=0

(rt+j − r) .

Suppose the central bank engineers today an increase of 1pp in the short rate rt, resultingin an immediate fall in xt by σ1 and a move in πt by κσ1. On the other hand, if the interestrate increase occurs at some date in the future, the response is larger. For example, anincrease promised in the infinite future would have an immediate contemporaneous effect oninflation of κσ1/ (1− β). If the discount factor β = 0.99, this response would be 100 timesgreater than the response to a contemporaneous increase.

Second, if the nominal rate instead of the real rate is kept constant before the shock, thereis another amplification channel with inflation feeding back to the real rate. The inflationresponse at the date of the shock in the future is also negative, but the closer the current date,the larger the inflation response, as the output gap is expected to remain negative. Thenconsumption continues to decrease when solving backward – from the date of the shock tothe current date – as the real interest rate rises, driven by falling inflation. This will amplifythe negative response of inflation even further. In order to overcome this puzzle, a standardDSGE model has to be modified such that an additional discount parameter appears in frontof the short run interest rate.96

As shown below with a simulation exercise, we do not observe such a puzzle in FR-BDF.The reason for this comes from the real side rather than the nominal one, since in FR-BDFprices and wages are modeled in a similar manner to the DSGE approach, i.e. assumingprice and wage stickiness. While FR-BDF does not have an explicit Euler equation, it doeshave an equation relating current desired consumption to permanent income yH,t and thepremium of the real household bank lending rate rLH,t over the trend of the real short ratert:

ct = α + yH,t − σ2 (rLH,t − rt) (145)96Such modifications include e.g. heterogenous agents (McKay et al., 2016), a weakly credible central bank

(Coenen & Wieland, 2004), overlapping generations (Del Negro et al., 2012) and finite planning horizon(Gabaix, 2016).

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The key conceptual difference between the two demand curves – (143) and (145) – is thatin FR-BDF consumption depends on the household lending premium instead of the shortrate. The former is constructed using the 10-year government bond, which itself depends ona discounted term structure equation – i.e. an equation describing the 10-year rate with aterm premium and a discounted sum of future short rates Rt+j. Because of this discountedterm structure, the later the announced shock occurs, the smaller the effect of the futureshort run interest rate on the household bank lending rate. In addition, (145) departs fromassumptions regarding the parametrization made in standard DSGEs by assuming that thediscount factor used to construct the permanent income term yH,t (0.95 or equivalently anannual discount rate of 25%) is low compared to values consistent with observed interestrates97 and that the risk aversion term implied by the term σ2 ≈ 0.55 is at a high value,around 10, compared to estimates of the DSGE literature, which generally lie in a rangegoing from 1 to 5.

6.3.2 Quantitative comparison

Figure 6.3.1: Shock to the short run interest rate (annualized, in pp)

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In order to illustrate the point, we present simulations of FR-BDF under model-consistentexpectations and a fully exogenized euro area in which the annualized interest rate is loweredby 25bp with respect to its steady state for one to eight periods respectively, see Figure 6.3.1.The short run interest rate follows a simplified AR(1) rule to avoid complications related tointeractions with the euro area. We also assume that there are no changes in the patternsof intra euro area trade, either in volumes or prices. Real GDP growth and consumer priceinflation are computed as year on year changes.

The results of the exercises are shown in Figure 6.3.2. Comparing all models, two resultsstand out. First, for a standard monetary policy shock – a shock that lasts only one quarterand contains no forward guidance – the peak output response is weaker in FR-BDF than in allthe DSGE models considered (responses from FR-BDF are plotted on the right-hand axis).The same remark applies for inflation although the model with Finite Planning Horizons(FPH) delivers a peak impact that resembles that of FR-BDF.

97This is due to attempts to approximate terms relating to uncertainty and risk aversion. See Section 4.6.1or Reifschneider (1996) for details.

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Second, FR-BDF does not suffer from the forward guidance puzzle, and non-baselineDSGE models provide either a partial or full solution to the puzzle.98 In FR-BDF, the peakimpacts on GDP and inflation are almost perfectly linear in the duration of the forward-guidance announcement. Among DSGE models, all variants of the baseline DSGE modelattenuate the response to forward-guidance announcements, although the additional dis-counting only partially solves the puzzle. In FPH, the impacts are even concave in thehorizon of the announcement: the effect of forward guidance fades with the horizon.

These results are sensitive to specific details relating to the specification and parametriza-tion of the models. In FR-BDF, agents are implicitly characterized by a very high degree ofrisk aversion. In most DSGEs this elasticity is calibrated to values close to 1. In unreportedsimulations, the peak effects of output and inflation in the FPH model are virtually identicalto those of FR-BDF if it is calibrated with a risk aversion coefficient ten times as large as inthe baseline calibration. In this case, the elasticity of output to the real interest rate has asimilar magnitude in FPH as in FR-BDF.

Figure 6.3.2: Responses of GDP and Inflation to Forward Guidance Policies

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Note: The figures plot the peak effect on GDP (in percentage points deviation from steady-state)and year-on-year inflation (in percentage points) of the announcement that interest rates willbe 25bp lower for n = 1, ..., 8 quarters. Non-discounted NK is a simple DSGE model subjectto the forward guidance puzzle. Discounted NK is the DSGE model that embeds the idea ofcredit-constrained households, as suggested in McKay et al. (2017). FPH is the DSGE model ofWoodford & Xie (2019) with a finite planning horizon set on average to 4 quarters.

98Note that our characterization of the absence of the puzzle, i.e. a linearity of peak responses withrespect to durations of shocks instead of an exponential curvature, is somewhat different from that foundin the academic literature, where the puzzle is analyzed by studying the dependence of the magnitude ofcontemporaneous effects on the how far in the future is the date of the actual policy change, which istypically assumed to last only for a single period. In this context the puzzle concerns merely the fact thatthe magnitude is increasing. The structure of our experiment and the characterization was chosen due toits closer proximity to actual policy applications, where central banks promise to keep the interest ratesconstant for some specific period.

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6.4 Asset purchase programmes99

With interest rates at the effective lower bound, the standard monetary policy tools havereached their limit within the euro area. In their place, the Eurosystem has applied sev-eral unconventional tools, including a variety of asset purchase programmes (APP). Theseprogrammes aim to provide monetary accommodation by transferring liquidity directly tomarket participants in exchange for various assets, including government bonds, thus by-passing the traditional lending and money creation channels. The empirical literature hasidentified effects on not only term premia, but also on exchange rates.

We use the estimated response of financial variables to study the effects of these policieson the French economy between 2015 and 2018 via simulations of FR-BDF. To do so, weconstruct a counterfactual outcome by removing the effects of APP on the term premium andexchange rates. The simulations are conducted both under VAR-based and model-consistentexpectations. In the main experiments we assume that France is economically disconnectedfrom the rest of the euro area, e.g. there are no changes in foreign prices or foreign demandfrom the rest of the euro area.

To obtain an assessment of the effects of APP on the French economy we use a sequenceof shocks to the term premium and to the nominal exchange rate.100 The shocks on termpremium are constructed based on a total effect of 100bp estimated by Eser et al. (2019).This total effect is then divided into four parts (shocks) consistent with the weights of theEurosystem asset purchase packages, see Table 6.4.1. The first package is treated in a specificway, as it was partially anticipated in 2014Q4 and detailed in 2015Q1. In this case we assumean ad hoc repartition of 1/3 and 2/3 between the quarters.

We set the total effect on the effective exchange rate to 9%. This number comes froma 12%-effect on the euro-dollar exchange rate estimated by Dedola et al. (2018) and theassumption that other European currencies (except the British pound) remain pegged to theeuro. We then divide it into four shocks using the same method, see Table 6.4.1.

Table 6.4.1: Eurosystem asset purchase packages and their effects on the term premiumand the nominal exchange rate

Package Amount Term prem. effect Exch. rate effect

January 2015 1140 -21.8 in 2014Q4 -1.9 in 2014Q4-43.7 in 2015Q1 -3.8 in 2015Q1

December 2015 360 -20.7 in 2015Q4 -1.8 in 2015Q4March 2016 240 -13.8 in 2016Q1 -1.2 in 2016Q1

Total 1740 -100 -9Note: Package amounts measured in billions of euros. Effects of shocksare annualized basis points. Effects realized in 2014Q4 are assumedto be anticipatory for the package of January 2015.

99We would like to thank Stephane Dees for supplying the data on shocks, for clarifying their constructionand for offering helpful comments.

100We omit explicit modeling of effects on stock prices, as FR-BDF accounts for such movements implicitlyby modeling cost of equity via the long-run rate which itself is affected by the term premium.

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Our main results under VAR-based expectations – presented in the first row of Table6.4.2 and in Figure 6.4.1 – indicate that the APP had notable real and nominal effects onthe French economy.101 The table shows yearly inflation and GDP growth rates which aresubstantial and positive. Overall, APP boosts investment and exports through a fall inlong interest rates and a depreciating exchange rate. On the nominal side, it first results inimported inflation and then in domestic inflation due to a rise in factor costs. The second andthird rows of Table 6.4.2 show simulation results from experiments where it is assumed thatthe APP affect only the term premium and exchange rates – the movements in the nominaland real side of the economy mainly appear to be the result of movements in exchange rates.

In contrast, under MCE, we find that in terms of the GDP growth rate the difference inthe short run is not very striking, although inflation reacts more strongly as expectationsregarding positive economic developments in the future result in a more pronounced Phillipseffect today. Effects on investment are accentuated but simultaneously, households expect tolose some purchasing power and thus do not raise their consumption as they would otherwise.

The importance of this inflationary effect is also demonstrated in the fifth and sixth rowsof Table 6.4.2. First, under MCE the inflationary effect of exchange rate movements is muchlarger. Furthermore, note that under MCE, the share of the total real effects arising fromthe exchange rate channel is smaller than under VAR-based expectations, and that in thelong run GDP growth turns negative. This is due to the faster adjustment of the economyin comparison to the VAR case. This leads to a reduction in purchasing power with negativeconsequences in the medium run on consumption. In addition, competitiveness reverts toits pre-shock level as a result of domestic inflation. Note that in the VAR case these effectswill also eventually lead to negative GDP growth.

Finally, two additional experiments were conducted in the same framework. In the firstwe investigated the importance of the assumed dynamics of the persistence of the shock.In this robustness check we instead assume that the shock persists only for a single quarterand then revert with its historical persistence. We find that when agents form their futurebeliefs using the VAR framework, the differences – with respect to responses obtained underthe baseline assumptions regarding the persistence of the shocks – are at the start of thesimulation relatively minor, with e.g. the peak effect on year-on-year GDP growth beingonly 0.02pp larger. At the end of the simulation period the differences increase, so thatthe difference in GDP growth rates is roughly -0.15pp and -0.1pp for inflation. Under theforward looking MCE framework the results are qualitatively similar for GDP, but stronger.For inflation this difference in assumptions plays a rather large role: under MCE the peakeffect is reduced by -0.25. At the end of simulation the difference in GDP growth rates isroughly -0.1; inflation is reduced by -0.3.

In the second experiment we studied the importance of Euro area dynamics. We carryout this exercise with the help of a satellite model, which is constructed using tables ofimpulse responses that result from the Eurosystem’s Basic Model Elasticity (BME) exercise.The tables describe the responses of each individual country to a variety of economic shocksin the models of each national central bank participating in the exercise; based on the tablesit is possible to conduct an analysis of the joint response of the euro area. We construct our

101Note that the slight fall in inflation at the start of 2016 shown in Figure 6.4.1 is explained by a basiseffect arising from the year-on-year computation of the growth rate.

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alternative experiment by applying the shocked deviations of real demand for French importsand the prices of foreign exports as additional innovations to the French economy.102 Bothof these variables exhibit a similar dynamic, where they at first increase and then revertstrongly after 2016.

We find that for output the cumulated response of the growth rate over the period 2015-2018 remains almost unchanged. However, the response is now more front-loaded, with agreater response occurring in 2015 and 2016, and conversely a smaller response occurring in2017 and 2018. In contrast to output, the response of inflation is unambiguously stronger,although not very much so, as the cumulated increase is just 0.06pp over the simulationperiod.

Figure 6.4.1: Year-on-year French output growth and inflation (percent), deviation frombaseline

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102The satellite model also produces euro area output and price level as outputs. Transformations of thesevariables appear in the expectations of FR-BDF. An alternative exercise was also conducted where the effectof these expectations shocks was taken into account. As the difference turned out to be very small, we omitthe results of this second exercise. Further details are available on request.

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Table 6.4.2: The effect of APP on average year-on-year inflation and GDP growth, percent

GDP growth Inflation2015 2016 2017 2018 Total 2015 2016 2017 2018 Total

VARFull 0.15 0.34 0.24 0.09 0.82 0.17 0.25 0.31 0.35 1.08TP 0.03 0.06 0.07 0.05 0.21 0.00 0.02 0.04 0.06 0.13ER 0.12 0.28 0.18 0.04 0.62 0.17 0.22 0.26 0.29 0.95

MCEFull 0.20 0.38 0.19 -0.04 0.73 0.27 0.52 0.59 0.48 1.85TP 0.08 0.14 0.10 0.02 0.35 0.04 0.12 0.16 0.12 0.44ER 0.12 0.24 0.08 -0.06 0.38 0.23 0.39 0.42 0.35 1.40

Note: "Full" describes results from the full experiment, "TP" results from an experimentwith a shock just to the term premium and "ER" a similar experiment with a shock tojust the exchange rates.

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7 ConclusionIn this paper, we have described in relative detail the features and applications of the newsemi-structural model of the Banque de France for the French economy, FR-BDF. We wouldlike to highlight here some of the key novelties in FR-BDF, particularly in comparison tothe old forecasting model, Mascotte.

As was initially intended for the project, FR-BDF has a rich set of financial channels;e.g. shocks to interest rates or other financial variables of the model have economicallymeaningful and realistic effects on the macroeconomy, both real and nominal. This is inparticular clearly demonstrated by the various experiments we have carried out for assessingmonetary transmission and is in stark contrast to Mascotte, in which monetary policy shockshave for example relatively modest effects on macroeconomic dynamics. On the other hand,it would seem that some of the issues related to monetary policy shocks present in textbookDSGEs – where the interdependence of interest rates and real activity is known to be verystrong – are not present in FR-BDF. FR-BDF does not suffer from the forward guidancepuzzle.

When analyzing the dynamics of FR-BDF, explicit expectations play a large role. Theexpectations model E-SAT describes a backward-looking method for achieving this role forexpectations, although FR-BDF can also be operated under model-consistent expectations.The economic impact of expectations can be seen from e.g. the various figures describingdynamic contributions. We would also like to emphasize the importance of how expectationsare modeled in FR-BDF: Eurosystem asset purchase programmes have for example strongereffects on inflation under MCE in comparison to backward-looking expectations.

Furthermore, the model has a well-defined long run in the sense that it has a balancedgrowth path toward which it converges in various simulations. This feature is completelyabsent from many semi-structural models used in forecasting.

However, further research and development is still required. Two key model features,which were outside the scope of the original project, are planned to be included in the model:connecting the model to the euro area and accounting in more detail for the interactionbetween the macroeconomy and the financial system.

Connecting the French economy to the euro area in FR-BDF is a two-fold process. Thefirst step consists of accounting for various indirect trade channels through which shocksaffecting the euro area would be passed on to France. The second requires modelling en-dogenous monetary policy – as it is, it is assumed that French economic developments haveno effect on the short rate. Given France’s rather large share in the euro area economy, thisis clearly a strong approximation on which it would be easy to improve, leading to gains inthe accuracy of the model, in particular policy analysis. This extension would be separatefrom the forecasting use of the model, where it is not applicable for reasons described insection 4.11; instead it would only be applied in policy analysis.

The central features regarding the interaction between the macroeconomy and financialmarkets that FR-BDF still lacks are the causal channel through which the financial sectoris affected by macroeconomic developments and the amplification that could arise from afinancial accelerator at least in times of financial stress. The former would require carefulmodelling of the dependence of e.g. house and stock prices on aggregate dynamics. Thelatter would require major improvements in the way in which credit and leverage dynamics

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are modeled in order to be able to relate them to the financial conditions faced by householdsand firms.

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A Appendix

A.1 Stability conditions, E-SAT model

In order to simplify the analysis, we assume here that the French output gap does not reactto that of the euro area (δq = 0).A.1 Without this feedback, the analysis of the dynamicof the system can be decomposed recursively into three blocks: AR(1) of the targets, theFrench economy and the euro area.

As the first block does not raise any issues, we start by looking at the second block, i.e.French variables Qt and πQ,t. Since the interest rate does not react to domestic variables,the Taylor rule does not play any role for ensuring the stability of this block. The stabilityof this block will depend on the 2×2 sub-matrix Hq,π, related to the dynamics of the outputgap and inflation:

Hq,π =

[λq σqκπ λπ

]The system is stable, if and only if the two eigenvalues of Hq,π − I are negative. In

the absence of any feedback of macro variables on the interest rate, stability occurs whenthe determinant Hq,π − I is positive and its trace is negativeA.2. These two conditions areequivalent to:

(1− λq)(1− λπ)− κπσq > 0

(1− λq) + (1− λπ) > 0

The first condition requires that the destabilizing role of the real interest rate (related toκπσq and λq) be sufficiently small compared to the anchoring degree of inflation (1 − λπ)and of the output gap (1 − λq). The second condition should be always verified, providedwe have some anchoring of inflation and the output gap (λπ < 1 and λq < 1).

Concerning the third block of the euro area, if there were no feedback from the Taylorrule (αi = βi = 0), stability conditions would be the same for parameters λq,ea, λπ, κπ,eaand σq,ea. With a feedback of monetary policy on macro variables, the stability should bereinforced and, hence, these stability conditions should be sufficient.

A.2 Implications for the term premium, E-SAT model

There are two types of bonds in FR-BDF: the classical one-period bond that returns RSt ,

and the long-term bond that returns RLt . The term premium is defined as the difference

between the return on the long-term bond and the expected return on the short-term one,where the latter is the expected sum of all future short-term rates over a horizon equal toA.1Our numerical simulations show that this channel does not change qualitatively the dynamics of E-SAT.A.2This condition only ensures stability of the output gap and inflation. In order to achieve stability of the

real exchange rate (here, pea/pQ), we could have considered adding a competitiveness channel, i.e. addingthis variable within the IS curve. However, (i) the identification of the parameter corresponding to thischannel is very weak; (ii) it would create either persistent responses or oscillations depending on the size ofthis parameter.

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the maturity of the corresponding long-term asset:

RLt − (1− pL)

∑∞z=0 p

LRSt+z.A.3

The term premium is unobserved and in order to compute it we approximate RLt with

the return on the euro area 10 year government bond, RSt is the 3-month Euribor and

pL = 1− 1−1/RL

1−1/( ˆRL)40, where RL = (1+RL

t /100). The expected return of the short-term bond is

computed by reversing the E-SATmodel, i.e. (1−pL)∑∞

z=0 pLRS

t+z = (1−pL)Zt[(I−pLH)−1]′,for the definitions of Z and H see equation 1.

We present the term premium consistent with our estimated VAR in the right-hand plotof Figure A.1. Starting from 2012, the term premium is negative, which contradicts theresults of the term structure model of the Banque de France. In order to bring our model inline with the official views of the Banque de France, we calibrate λı parameter to 0.985, seeleft-hand plot of Figure A.1.

Figure A.1: 10-year bond yield, expected nominal interest rate and term premium.

1995 2000 2005 2010 2015 2020-4

-2

0

2

4

6lambdaiibar=0.9 (estimated)

1995 2000 2005 2010 2015 2020-1

0

1

2

3

4

5

6lambdaiibar=0.985 (calibrated)

10-year sovereign bond rateExpectation componentTerm premium

A.3For detailed derivations see the note DEMFI_2016_032.

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A.3 Additional figures

Figure A.1: Prior (gray) and posterior distributions (black).

0 0.5 1

0

2

sigmaq_ea

0 0.5 1

0

5lambdapi

0 0.5 1

0

2

4

lambdapi_ea

0 0.5 1

0

5

10

kappapi

0 0.5 1

0

20

40

kappapi_ea

0.2 0.4 0.6 0.8 1

0

10

lambdaii

0 1 2 3

0

1

alphaii

0 0.2 0.4 0.6

0

5

10

betaii

0.1 0.2 0.3 0.4 0.5

0

10

sepsQhat

0.1 0.2 0.3 0.4 0.5

0

10

20sepspiQ

0.1 0.2 0.3 0.4 0.5

0

10

20

sepspiQ_ea

0.1 0.2 0.3 0.4 0.5

0

50

sepsii

0.2 0.4 0.6 0.8

0

10

sepsQhat_ea

0 0.5 1

0

10

deltaq

0.2 0.4 0.6 0.8 1

0

5

lambdaq

0.2 0.4 0.6 0.8 1

0

10

lambdaq_ea

0 0.5 1

0

2

4

sigmaq

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Figure A.2: GDP components’ responses to a monetary policy shock, see section 6.2

-.2

-.1

.0

.1

.2

.3

-.2

-.1

.0

.1

.2

.3

1 21 41 61 81

VAR_based Hyb.E. MCE

Quarters

Output

-.20

-.15

-.10

-.05

.00

.05

.10

.15

.20

.25

-.20

-.15

-.10

-.05

.00

.05

.10

.15

.20

.25

1 21 41 61 81

VAR_based Hyb.E. MCE

Quarters

HH consumption

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1 21 41 61 81

VAR_based Hyb.E. MCE

Quarters

Business investment

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1 21 41 61 81

VAR_based Hyb.E. MCE

Quarters

HH investment

-.6

-.4

-.2

.0

.2

.4

.6

.8

-.6

-.4

-.2

.0

.2

.4

.6

.8

1 21 41 61 81

VAR_based Hyb.E. MCE

Quarters

Exports

-.4

-.2

.0

.2

.4

.6

.8

-.4

-.2

.0

.2

.4

.6

.8

1 21 41 61 81

VAR_based Hyb.E. MCE

Quarters

Imports

139